CSS, /3:££S <3V NOAA Technical Report EDS 24 .< °' f°* Sf ATES 0« *" A Note on a Gamma Distribution Computer Program and Computer Produced Graphs Washington, D.C. May 1977 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration Environmental Data Service NOAA TECHNICAL REPORTS Environmental Data Service Series The Environmental Data Service (EDS) archives and disseminates a broad spectrum of environmental data gathered by the various components of NOAA and by the various cooperating agencies and activities throughout the world. The EDS is a "bank" of worldwide environmental data upon which the researcher may draw to study and analyze environmental phenomena and their impact upon commerce, agriculture, industry, aviation, and other activities of man. The EDS also conducts studies to put environmental phenomena and relations into proper historical and statistical perspective and to provide a basis for assessing changes in the natural environment brought about by man's activities. The EDS series of NOAA Technical Reports is a continuation of the former series, the Environmental Science Services Administration (ESSA) Technical Report, EDS. Reports in the series are available from the National Technical Information Service, U.S. Department of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22151. Price: $3.00 paper copy; $1.4~5 microfiche. When available, order by accession number shown in parentheses. ESSA Technical Reports EDS 1 Upper Wind Statistics of the Northern Western Hemisphere. Harold L. Crutcher and Don K. Halli- gan, April 1967. (PB-174-921) EDS 2 Direct and Inverse Tables of the Gamma Distribution. H. C. S. Thorn, April 1968. (PB-178-320) EDS 3 Standard Deviation of Monthly Average Temperature. H. C. S. Thorn, April 1968. (PB-178-309) EDS 4 Prediction of Movement and Intensity of Tropical Storms Over the Indian Seas During the October to December Season. P. Jagannathan and H. L. Crutcher, May 1968. (PB-178-497) EDS 5 An Application of the Gamma Distribution Function to Indian Rainfall. D. A. Mooley and H. L. Crutcher, August 1968. (PB- 180-056) EDS 6 Quantiles of Monthly Precipitation for Selected Stations in the Contiguous United States. H. C. S. Thorn and Ida B. Vestal, August 1968. (PB-180-057) EDS 7 A Comparison of Radiosonde Temperatures at the 100- , 80-, 50-, and 30-mb Levels. Harold L. Crutcher and Frank T. Quinlan, August 1968. (PB-180-058) EDS 8 Characteristics and Probabilities of Precipitation in China. Augustine Y. M. Yao, September 1969. (PB-188-420) EDS 9 Markov Chain Models for Probabilities of Hot and Cool Days Sequences and Hot Spells in Nevada. Clarence M. Sakamoto, March 1970. (PB-193-221) NOAA Technical Reports EDS 10 BOMEX Temporary Archive Description of Available Data. Terry de la Moriniere, January 1972. (COM-72-50289) EDS 11 A Note on a Gamma Distribution Computer Program and Graph Paper. Harold L. Crutcher, Gerald L. Barger, and Grady F. McKay, April 1973. (COM-73-11401) EDS 12 BOMEX Permanent Archive: Description of Data. Center for Experiment Design and Data Analysis, May 1975. EDS 13 Precipitation Analysis for BOMEX Period III. M. D. Hudlow and W. D. Scherer, September 1975. (PB-246-870) EDS 14 IFYGL Rawinsonde System: Description of Archived Data. Sandra M. Hoexter, May 1976. (PB-258-057) EDS 15 IFYGL Physical Data Collection System: Description of Archived Data. Jack Foreman, September 1976. (PB-261-829) (Continued on inside back cover) ATMOSal NOAA Technical Report EDS 24 A Note on a Gamma Distribution Computer Program and Computer Produced Graphs Harold L. Crutcher, Grady F. McKay, and Danny C. Fulbright National Climatic Center Asheville, N.C. May 1977 U.S. DEPARTMENT OF COMMERCE Juanita M. Kreps, Secretary National Oceanic and Atmospheric Administration Robert M. White, Administrator a u Environmental Data Service x Thomas S. Austin, Director 5 g. For sale by the Superintendent of Documents, U.S. Government Printing Office O Washington, D.C- 20402 to Stock No. 003-017-00400-7 ACKNOWLEDGMENTS Appreciation is expressed to Professor Hubert W. Lilliefors for his considerable help in the preparation of table 5 which has previously been published in the Journal of Applied Meteorology (Crutcher, 1975) and to the Journal for permission to duplicate the table. Acknowledgment is made to L. R. Shenton and K. 0. Bowman for permission to quote material . Appreciation is expressed to Professor H. K. Kao and the American Institute of Industrial Engineers for permission to use figure 5 from his article (Kao, 1968). Acknowledgment is made to Miss Jane Langford and Mr. Charles Tipton, Junior Fellows, for developmental programming, to Mr. Doug Swann for drafting work, to Dr. Nathaniel B. Guttman for review of the paper, and to Mrs. Margaret Larabee for manuscript typing. Appreciation also is expressed to the staff at EDS's Environmental Science Information Center, in particular to Mr. Patrick McHugh, for the care given to the editing of this paper. Mention of a commercial company or product does not constitute an endorse- ment by the NOAA Environmental Data Service. Use for publicity or advertising purposes of information from this publication concerning proprietary products or the tests of such products is not authorized. n CONTENTS Acknowledgments ii Li st of f i gures i v Symbols used in this report v Abstract 1 I. Introduction 1 II. The gamma distribution function 3 III. The general gamma distribution function 5 IV. Parameter estimation 6 V. Origin 7 VI. Procedures for small shape parameters ,. . . 12 VII. Procedures for large shape parameters 22 VIII. Mixed distributions 28 IX. Gamma distribution function computer program 29 X . Graphs 35 XI . Graph paper 44 XII. Future modifications to the program 48 References 49 Author i ndex 55 Appendix 1 . Tables Appendix 2. Use of algorithm to determine quantile ratio for a given shape parameter and probability. Appendix 3. FORTRAN IV electronic computer program for application of the gamma distribution function to data sets and computer output to microfilm graphs. in FIGURES Figure 1. Selected gamma distribution function curves... 11 Figure 2. Precipitation probabilities for Auburn, AL, during the first week of the climatological year. March 1-7, 1930-1969 30 Figure 3. Precipitation probabilities for Auburn, AL, during the second week of the climatological year. March 8-14, 1 930- 1 969 31 Figure 4. Auburn, Alabama, precipitation probabilities March 1-7, 1 930-1969 36 Figure 5. Auburn, Alabama, precipitation probabilities March 8-14, 1 930- 1 969 37 Figure 6. Auburn, Alabama, precipitation probabilities March 1-7, 1 930-1 969 42 Figure 7. Gamma plot of failure-age (after Kao, 1968) 46 IV Symbols used in this report a (1) gamma distribution variable dependent on second and third moments (2) constant b (1) plotting parameter equal to c (2) constant c (1) plotting parameter with a default option to 0.44 (2) constant d Constant e Exponential; 2.7183 f Function f. Specified constants g (1) function (2) as subscript, with respect to gamma distribution h Geometrical constant i (1 ) sample number (2) subscript j (1 ) sample number (2) subscript k Subscript, such as i or j In Natural logarithm m (1) number (2) number of iterations n (1 ) number of data (2) number of iterations, terms p (1) probability of nonzero amounts; NX/NNX (2) probability level q (1) (1 - p), probability of zero amounts (NNX-NX)/NNX (2) constant r Variable, reliability index s Variable t Variable t Average t; the overbar indicates an averaging process dt Derivative of t v Variable dv Derivative of v x Variable, here generally y - a dx Derivative of x x' Transformed x, as (y - a)/3 x Average x, nonzero amounts only y Variable of function y Average y, nonzero amounts only z Variable of integration A Numbers (Bernoulli) D Determinant F Function G(x) Gamma distribution function for a measured set excluding zeros standard cumulative distribution function (cdf) G" 1 Inverse H(x) Gamma distribution function for a measured set including zeros I Sample number J (1) number of duration periods (2) number of data combined K Kolmogorov (1933) K-S Kolmogorov-Smirnov M Moment; subscripts indicate type of moment ML Maximum likelihood NX Number of data excluding zeros NNX Number of data including zeros P Probability P , Probability with respect to one tail normal distribution P „ Probability with respect to two tail normal distribution n2 J r Q Constant R Reliability function S Smirnov (1936, 1948) X Untransformed variable (i.e., an original datum) Y Untransformed variable (i.e., an original datum) VI a Alpha; (1) origin (2) probability level for rejection 3 Beta; scale parameter Y Gamma; shape parameter 3 Beta hat; maximum likelihood estimate of sample scale parameter Y Gamma hat; maximum likelihood estimate of sample shape parameter 6 Beta star; Thorn's (1958, 1968) estimate of scale parameter Y Gamma star; Thorn's (1958, 1968) estimate of shape parameter e Epsilon; error z, Zeta; median, variable y Mu; mean p Rho; reliable life a Sigma; standard deviation a 2 Sigma square; variance t Tau; quantile dx Derivative of x Phi; function, standard normal distribution x Chi x 2 Chi -square r Gamma; gamma function a Delta; increment z Sigma; summation Equal to = Identically equal to / Sigma; Integral > Greater than » Much greater than >_ Equal to or greater than < Less than <_ Equal to or less than Multiplication sign Overbar; averaging process °° Infinity vn A NOTE ON A GAMMA DISTRIBUTION COMPUTER PROGRAM AND COMPUTER PRODUCED GRAPHS Harold L. Crutcher, Grady F. McKay, and Danny C. Ful bright National Climatic Center, Environmental Data Service, NOAA Asheville, N.C. ABSTRACT. The gamma distribution function may be used as a model for many sets of data. The electronic computer program for this function in the Formula Translator (FORTRAN) IV (1) provides the analytic solution to a set of data, (2) gives the probabilities of exceeding or not exceeding arbitrary amounts, (3) indicates the amounts exceeded or not exceeded for arbitrary probabilities, and (4) provides for computer output to microfilm, drum, or flatbed plotters. The program, in its general form, permits a maximum of 52 entries which will suffice for those dealing with weekly data through the year. In addition, such as in weekly precipitation studies, the user has the option to compute the two- and three-week duration period distri- butions in one pass of the data. These computations are done without program change but by appropriate changes in the control cards. This feature is not limited to the study of precipitation data nor to the time intervals used. An option permits the computation of the required proba- bilities and inverses when only the scale and shape parameters are given. By appropriate modification, where only one set of data is being examined, the program can accept a string of input data longer than 52. I. INTRODUCTON This report presents, for the gamma distribution model, (1) an improved electronic computer program in the Formula Translator (Fortran) IV to provide an analytic solution for a data set and (2) computer produced graphs which permit better assessment of the model's applicability. The five modifications planned for the computer program presented in the previous report, NOAA TR EDS 11 [Crutcher et al . (1973)], were (1) a subroutine for the determination of an acceptable location (origin) parameter, 1 (2) a possible subroutine for the debiasing of the maximum likelihood and Thorn (1958) shape and scale estimators, (3) modification of routines to permit calculation of probabilities for shape parameter values, plots, and other x-y type plotters using linear scale plotting, (4) a subroutipe for cathode ray computer output plots, and (5) a separate program designed for low values of the shape parameter (i.e., y 1 1.000). The present paper presents the last three modifications as well as the development of an algorithm to extend computing capabilities to shape parameter estimates of about 100. Both the extension of capability to the lower and to the higher values provide the serendipitous dividend of re- ducing the internal computing time (cpu) more than 50 percent. N0AA TR EDS 11 provides examples and work sheets for gamma graph paper. One other form of graph paper is illustrated in section XI. In the present paper, output plotter and computer output-to-microfilm routines will provide some graphs with plotted data. These routines may be used to obtain any needed graph paper which may then be duplicated as requested. The program, in its general form, permits a maximum of 52 entries which will suffice for those dealing with weekly data through the year. In addition, in precipitation studies, the user has the option to compute the two- and three-week duration period distributions in one pass of the data. These computations are done by appropriate changes in the control cards. This feature is not limited to the study of precipitation data. An option permits the computation of the required probabilities and inverses when only the scale and shape parameters are given. By appropriate modification, where only one set of data is being examined, the program can accept a string of input data longer than 52. II. THE GAMMA DISTRIBUTION FUNCTION Many processes produce data distributions that the gamma distribution model describes well. Naturally, considerable literature exists for this distribution. The model serves for reliability life tests and fatigue problems. It offers advantages in the study of many multiple component systems where time-to-failure is an important feature. There are many other applications. The generalized gamma distribution contains a family of specialized dis- tributions. These cover a wide variety in form and usefulness. Kao (1968) presents a good discussion. As indicated by Kao and others, the gamma dis- tribution may be called the "rope" distribution. A rope does not break until the last fibre has parted. The gamma distribution describing the total of all the fibers of a rope has a shape parameter for the breaking which is the sum of the parameters for the individual fibers or subsets of fibers. In a similar way, other physical processes may be described by the gamma distribution. Haggard, Bilton and Crutcher (1971) discuss such a set which consists of storm processes which produce rain. The specific subset consists of hurricanes which cross the 1 ,000-f t contour of the Appalachians. Each hurricane consists of entrainment and coalescence of sub-storms. The gamma distribution is one of the models which is used in reliability techniques when the "rope" philosophy is employed. As presented in NOAA TR EDS 11, the gamma distribution is a form of the Pearson Type III, Pearson (1916). Another form is the Wei bull which also can be used in reliability problems. Although not discussed further here, it is useful when the problem is of the "chain" type, analogous to a chain being no stronger than the weakest link. Pearson (1916) derives the gamma density function (Pearson's Type III) as the solution of a differential equation. The tables edited by Pearson (1922), with subsequent revision through 1957, and those by Pearson and Hartley (1954) permit application of the gamma distribution model to fit and graduate skew data. The above tables permit interpolation for fractional degrees of freedom for the chi-square distribution. Campbell (1923) provides perhaps the first tabulation of the inverse gamma function if only for integer values. These, of course, are equivalent to the chi-square distri- bution with integer degrees of freedom equal to twice the gamma values. Salvosa (1930) also provides useful tables. Cohen et al . (1969) extend the tables of Salvosa. Birnbaum and Saunders (1958) derive and use the gamma distribution as one of the models for material life length, which may be likened to the life of a storm or the time-to-failure of precipitation generating processes. Harter (1964, 1969) provides an excellent discussion and extends Pearson's tables. Yet, as Harter says, Pearson's work has no serious contender. III. THE GENERAL GAMMA DISTRIBUTION FUNCTION The general gamma distribution with origin parameter a(- °° < a < + °°), scale parameter 3(3 > 0), and shape parameter y(y > 0) has the probability density function shown in f(y;a,3,Y) = 3* Y (r(y))- 1 (y-a) Y ' 1 e"^" a ^ /3 ; y > a; - « < a ,y < + - and f(y;a,3,y) = 0, y <_ a. (1) The distribution function given by F(y;a,3,y) = // f(t;a,3,y)dt (2) is for all y > a. Fisher (1922) first develops the maximum likelihood (ML) equation for the solution for the incomplete gamma distribution known commonly as the gamma distribution. It is incomplete in the sense that the integral limits of the function do not range from - » to + °° but from some finite point such as a to + k where k is some real number. If the origin parameter a is zero, this distribution is a special case of the Pearson Type III distribution. The solution of the ML equation as developed by Fisher is difficult. Therefore, Thorn (1947) develops approximate solutions. Chapman (1956), Greenwood and Durand (1960), Gupta (1960), and Wilk et al . (1962) provide methods to esti- mate the gamma distribution parameters. Mooley and Crutcher (1968) discuss the variability of the parameter estimates of two gamma distributions. Schickedanz and Krause (1970) present tests for the scale parameters. Thorn's work leads to fruitful use of the gamma distribution in meteorologi- cal, climatological , and hydrological applications. Barger and Thorn (1949) furnish an evaluation of drought hazard. Friedman and Janes (1957) provide an estimation of rainfall probabilities. Thorn (1958) presents a note on the gamma distribution. Barger et al . (1959) give the chances of receiving selected amounts of n-week precipitation in the north-central region of the United States. The last is the model for a number of subsequent publications Hartley and Lewish (1959) manage the computer hardware and software for the above study. Thorn and Vestal (1968) provide a study of monthly rainfall in the conterminous United States. IV. PARAMETER ESTIMATION The gamma distribution (Pearson's Type III) includes the chi -square and the exponential distributions as special cases. Pearson (1922), Thorn (1958), and Hahn and Shapiro (1968) discuss this. Most statistical texts briefly discuss this also. The shape parameter y is equal to one-half the degrees of freedom for the chi-square distribution and is equal to 1 for the exponential distribution, while the scale parameter 3 is equal to 1 in the standardized case as well as the last two cases. Barger et al . (1959) provide plotting paper where the arguments are the mean and the variate. Overlaid straight lines represent probabilities. Each value of the shape parameter y requires a separate graph. In the same paper, Thorn's distribution curves, also prepared from Pearson's tables in 1957, appear. The probability and the variate divided by the scale parameter are the arguments with the shape parameter being overlaid in curved lines over the argument plot. Wilk et al . (1962) provide techniques to estimate the scale and shape parameters, and they indicate that computer routines are available to provide graphical plots in terms of the quantile probabilities of the distribution and scale units. The theoretical line of best fit is then a straight line. These authors provide a brief set of tables that allows the person with a desk calculator, slide rule, or paper and pencil to interpolate required probability values and scale values and to make a plot of the data against the line obtained from the estimate of the scale and shape parameters. Roy et al . (1971) incorporate the above paper. Thorn (1968) presents direct and inverse tables of the gamma distribution Thorn's tables fill in areas not covered by the Wilk et al . (1962) tables and repeat other portions for verification. Crutcher et al . (1973) provide graphical techniques to estimate the scale and shape parameters. V. ORIGIN The origin or location parameter a in equation (1) usually is set to zero However, there are cases where the origin is not zero. Elderton (1953) uses Pearson's moments to locate an origin from which the other parameters of the distribution may be measured. The necessary statements follow: a = origin = mode - a, a = (2M2/M 3 ) - (M 3 /2M 2 ), mode = t - (M 3 /2M 2 ), and origin = t - ((2M2)/M ? ) (3) where M and M 3 are the second and third moments from the mean of the distribution. Barger (1964) discusses this. The expression t - (2M^/M ) does not ensure a positive location estimate even though the observed values are all positive. In some cases, the estimate may be higher than the lowest observed and recorded value in the data set. Pitman's (1938) estimator for the location (origin) parameter is a minimum variance unbiased estimator if the scale and shape parameters are known. These parameters usually are not known and must be estimated. Pitman's technique is not examined further in this report. Hastings (1955) provides an equation for the estimation of the origin. Greenwood and Durand (1960) also study the estimation of the location parameter. Chapman (1956) provides a tabular aid for iterative procedures to solve for the origin parameter in the untruncated case. He indicates an additional procedure for the truncated case, providing there is sufficient supplementary information. These iterative procedures are not examined in this report. Blischke (1975, 1971, and in prior studies) pursues the solution to the problem. Blischke encountered the same difficulties in the estimation process as is mentioned for the Elderton estimator. This, of course, blocks the calculation of maximum likelihood estimators for the shape and scale parameters or in any estimation process where logarithms are used. Blischke suggests that the lowest value be used as the origin where the estimated origin is above the lowest observed datum. This is the maximum likelihood estimator for the origin. In NOAA TR EDS 11, Crutcher et al . (1973) found that fit to the gamma distribution may be rejected when this is done, even though a value slightly lower than the datum as the estimator for the origin is used. Blischke (1975) brings the work on estimation of the origin parameter up to date. The program presents several options for the origin. The default option uses zero as the origin. Such a case would be zero for measured precipita- tion. If prior experience or theoretical considerations indicate the va'lue(s) of the origin parameter(s), this option is entered in a control card that replaces the default option. A third option uses the lowest value less a small amount to ensure the positive number needed for the logarithms used in the maximum likelihood or Thorn's estimators. Additionally, if the lowest value occurs more than once, this value becomes the origin. The program processes the mixed distribution that consists of two cate- gories, the lower bound and the values above the bound. Categories by non- occurrence, such as zero precipitation, and the distribution of measurable precipitation above the bound is such a mixed distribution. Regarding the bias in the estimators of maximum likelihood, it is of interest to refer to Blischke' s work and to that of Shenton and Bowman (1970), Fisher (1922), and Thorn (1957, 1958). Here we reproduce the comments of Shenton and Bowman: "In this note we show that Thorn's statistics are: a) slightly biased, no matter how large the sample; however this bias is almost negligible for y>>0, and indeed is only of any real importance if y is small (say less than 0.1 approximately); the bias in finite samples is about the same as for the maximum likelihood estimators; 8 b) superior to the maximum likelihood estimators because their variances are less in large sample theory; there is evidence that this property holds in finite samples also; c) about as near to normality (as measured by skew- ness and kurtosis) as the maximum likelihood estimators; actually the distribution of 3 is generally nearer to the normal form than that of 3." * In the above, the 3 and 3 are respectively the maximum likelihood and Thorn estimators of scale parameters. Removal of bias in the estimators is not attempted in this program and report. Woodward and Gray (1975) discuss for the scale parameter the "Minimum Variance Unbiased Estimation in the Gamma Distribution," the "UMVU," through the use of a new infinite series. This technique is not examined in the present paper. Anderson and Ray (1975) present modifications designed to reduce the bias and mean square error for the two-parameter gamma distribution. Such will be examined later. In view of the large variability of the estimates (Andrews and Barger, 1956; Mooley and Crutcher, 1968), removal of the bias may or may not be appropriate. With the origin a obtained, the following expression is pertinent: x = y - a. (4) Then (1) becomes f(x;0,3,y) = 3" Y (r( Y ))" 1 x Y " 1 e* x/6 , < x < » and f(x;0,3,y) = x £ 0. (5) Thorn (1968 and in his earlier papers) utilizes this form. As shown by Thorn (1958) and by Wilk et al . (1962), if the variate x assumes a transform by division of the scale parameter 3, the distribution function develops as F(x';0,l,y) = (r(y))- 1 !*' jY-1 e -t dTj ^ > Q and F(x';0,1,y) =0, x' < , (6) that is a standard form with a = and 3 = 1 and is positive when x > 0. Figure 1 provides a picture of the effect of the shape parameter and scale parameter on the function curves. Here, the standardized scale (frequency) is plotted against the quantile x. Curves for shape parameters (A) 0.5, 1.0, 1.5, and 2.0, (B) 1, 2, 3, and 4, and (C) 1, 5, 10, 20, and 30 illustrate the effect. The shape parameter for 1.0 is shown on each subset, but the hori- zontal scale has been compressed. Hahn and Shapiro (1968) and Falls (1971) provide illustrations for other combinations of the scale and shape parameters. Reference to x 2 curves also may be made. Where y = 1> this is the same as the exponential distribution and the x 2 with two degrees of freedom. Stated somewhat differently, the random variable (l/2)x 2 with 2y degrees of freedom 2Y has the gamma density function with the scale parameter equal to one and the shape parameter equal to y. Wilk et al . (1962) and Thorn (1968) present the numerical methods to obtain the estimates of the gamma distribution scale and shape parameters £ and y- Masuyama and Kuroiwa (1951) provide a table for the likelihood solution of the gamma distribution. Those papers provide more detail. As Barger et al . (1959) indicate, the estimates of the parameters are subject to rather large variations due to sampling and estimating errors. Mooley and Crutcher (1968) discuss the variance of the probabilities of exceeding stated amounts based on work of Andrews and Barger (1956). For a particular gamma variate distribution, the product of the shape and ** scale parameters equals the mean of the nonzero quantities. That is, &y = y. If Yi> y 2 »---»y n are independent gamma variates with shape parameters equal n to y, » Y 2 »---»Y n » then Y = z y. is a gamma variate with a shape parameter n 1-1 equal to £ y,- (Kenney and Keeping, 1951 and Lancaster, 1969). This pro- lyl vides a useful tool for combining parameter estimates, thereby reducing the computation that would be required if the original data sets were combined. The division of the mean of the total set by the new shape parameter estimate provides the new scale parameter estimate. An option is available in the computer program discussed below that permits the calculation of probabilities from the input value of the parameter estimates in lieu of entry of original data with subsequent calculation of the estimates. 10' A <+> 1.0 0.8 0.6 0.4 0.2 r 1 r" — y 1 5 10 (C) 20 y 30 1 r^*^-* 1 ^~- 10 20 T -r 30 40 FIGURE 1. SELECTED GAMMA DISTRIBUTION FUNCTION CURVES. 11 VI. PROCEDURES FOR SMALL SHAPE PARAMETERS 1 . General One of the difficulties encountered in the computer program developed for NOAA TR EDS 11 was the increasing failure rate to converge as shape parameters, when less than 1, became less and less. Such failure rate also was noted as the shape parameter, when greater than 40, became larger and larger. Occa- sional intermittent failure at isolated points at higher and lower probability levels also occurred. This last is not extremely important for sufficient nearby points are available to permit easy interpolation. This section and the next section, incorporated in the new program developed and presented in this paper, resolve all three difficulties. But more than that, the internal computing time is decreased by more than 50 percent. These developments are extensions of the Williams (1946) and Pinkham (1962) techniques. As no direct inversion exists for the gamma distribution, the extraction of quantiles presents problems. Approximation techniques have their associated difficulties. In general, as the shape parameter decreases, the determination of quantiles becomes increasingly more difficult. Thorn (1968) gives tables of quantiles which were derived using approximation methods. He also recognized the difficulty of qUantile determination for small values of the shape parameter. Pinkham (1962), following Williams (1946), presented an approximation for the evaluation of the gamma probability integral which used the normal dis- tribution. Using the inverse of the Williams-Pinkham techniques, tables of conversion factors have been determined whereby, with few exceptions, quan- tiles can be computed for 52 selected probabilities for y = 0.1[0.1]4.4. In addition, these conversion factors are given for y = 0.01 to satisfy boundary requirements. The number 52 is arbitrary and was chosen to fit a sequence of 52 weekly values for a year. For the region of the tables for which conversion factors yield unreliable results, another approximation has been used for quantile determination. In essence, this section is an extension of the Williams-Pinkham techniques. Conversion factors for shape parameters 12 other than those given may be determined using an interpolating parabola. These values are given in table 1 of appendix 1. Tables of quantiles computed for 21 probability levels and probabilities for selected quantiles are presented in tables 2 and 3 of appendix 1. These are given for y = 0.01[0.01]0.10, 0.10[0.05]1 .0, and 1.0[0.1]1.5. These, over parts of the ranges, duplicate those presented by Wilk et al . (1962) and Thorn (1968). 2. The gamma distribution For continuity purposes, equations (1) and (6) are repeated here. The probability density function for the gamma distribution is f(y;a,3,y) = 3" Y (r( Y ))" 1 (y-a)^ 1 e- (y " a)/3 , y > a >.0 (1) and f(y;a,3,y) = 0, y <_ a , where y denotes the random variable, a is the location parameter (generally a = 0), 3 is the scale parameter, and y is the shape parameter. r(y) is the gamma function where y» the shape parameter, is used as the argument. Letting a = and defining x' = (y-a)/e, the distribution function becomes F(x';0,l, Y ) = (r(y))" 1 /*'" t^V^t, x' > (6) o and F(x';0,l,y) = 0, x' <_ , where x is the quantile and x' is a transformed variate. While the evaluation of (6) may be done using Pearson's expansion (1957), the evaluation may also be done using an approximation given by Pinkham (1962). Following Williams (1946), Pinkham developed an approximation to the gamma distribution probability integral (6) which entails the use of the standard normal distribution. Modifying Pinkham's notation, we can write I" 1 / t T Y "V T dT - [^./rn 2 Y o where h is a geometrical constant, t = x (r(y))' 1 r T^V- T dT « [P(h/t)] 2Y , (7) o P(h/t) = (v^T) -l +h/t e" z2/2 dz , (8) -hv/t 13 and where z replaces x as a variable of integration. Pinkham provides tables of h values for selected shape parameters and probabilities. 3. The inverse of the gamma distribution The use of Newton's Approximation to evaluate the inverse gamma distribu- tion has been discussed by Thorn (1968). An examination of (7) suggests that, given h values, quantiles of the gamma distribution may be derived by use of the inverse normal distribution function. For convenience, the following expressions are redefined as P g (t) = F(t;y) and P n2 (x) e P(h/t) . Now consider the approximation (7) to be P g (t) - [P n2 (x)] 2Y and take the (l/2y) root of both sides. This yields [P g (t)] l/2Y ~~ P n2 (x) . (9) Rewriting the right side of (9) in terms of a one- tail normal distribution gives (P n2 U) + l)/2. To maintain the equality expressed in (9), the left side is treated in the same manner. Defining P n ,(x) = (P n2 ( x ) + l)/2» expression (9) may be written as ([P g (t)] 1/2Y + D/2 « P ni (x) , (10) P nl (x) = (^T) where -l e" z2/2 dz. (11) Now x s h/t can now be computed by inverting (11). Given h values for the specified shape parameter, the quantile t may be determined in a straight- forward manner, i.e., t = (x/h) . Furthermore, if one is willing to accept t as an initial estimate of the quantile, an appropriate approximation technique improves the estimate of the quantile. For quantiles presented in appendix 1, the Newton Approximation substantially improves quantile estimates obtained by inverting (11). 14 4. Determination of h values By using Pinkham's approximation in conjunction with quantiles of the gamma distribution (Wilk, Gnanadesikan, and Huyett, 1962), tables of h values have been computed for 52 selected probabilities for y = 0.1[0.1]4.4 and for a lower boundary y = 0.01. Table 1 presents these h values as an extension to those of Pinkham. These values were determined by "guessing" an h, com- puting a quantile, and subsequently a probability for the quantile. The "guessed" h value was then incremented by some Ah, e.g., 0.1x10"* and the resulting h value was used to determine a new quantile from which the proba- bility again was computed. This incrementation and subsequent computation of quantiles and probabilities was continued until the relative error in the probability was at a minimum for the increment being used. The initial guess for each shape parameter was based primarily on Pinkham's values. Values of h for extremely low values of the shape parameter and extremely low probabilities should be used cautiously. In general, h values for those combinations of y and P (t) for which [P (t)] 1/2Y < 0.5xl0" 10 may yield un- reliable values of quantiles. Another method is given later which allows more reliable quantile computations. 5. Interpolation of h values For the determination of h values for shape parameters other than those presented in this paper, e.g., 0.336, an interpolating parabola (Southworth and Deleeuw, 1965) was found to yield h values of sufficient accuracy. The general equation of this parabola is h(y) = a lY 2 + a 2 y + a 3 , (12) where a , a , and a are constants to be evaluated. As three constants are to be determined, a system of three equations is required. This system may be obtained by sutstituting three functional values into the polynomial. The solution to any system of linear algebraic equations can be obtained using determinants and applying Cramer's Rule (Michal, 1947 and Perlis, 1952). Since 3rd order determinants are involved here, they may be evaluated by developing them by row or by column. If the order were higher, it would 15 be more advantageous to employ some other technique of evaluation to solve the system of equations. One such method would be the Gaussian Algorithm, which is a systematic elimination method. Let us examine the solution of a system of three equations, system be Let this ax+ax+ax=y 11 1 12 2 13 3 1 ax+ax+ax=y 21 1 22 2 23 3 2 ax+ax+ax=y 31 1 32 2 33 3 3 where a , a , a are the coefficients of unknowns x , x , and x and 11 12 33 12 3 y , y , and y denote function values. The determinant of the coefficients 1 2 3 is a a a 11 12 13 D = a a a 21 22 23 a a a 31 32 33 Solutions x , x , and x will exist provided that D f and the system is non- 12 3 homogeneous, i.e., y , y , and y f 0. 1 2 3 Three additional determinants are required to use Cramer's Rule with a system of three equations. These are D = i y a a 1 12 13 y a a 2 22 23 y a a 3 32 33 , D = 2 a y a 11 1 13 a y a 21 2 23 a y a 31 3 33 , and D = 3 a a y 11 12 1 a a y 21 22 2 a a y 31 32 3 By Cramer's Rule, the solutions are x = D D' 1 , x = D D" 1 , and x = D D" 1 . 11 2 2 3 3 The evaluation of the required determinants can be done easily by expansion 16 by minors as only 3rd order determinants are involved. Evaluation of D using the first row yields D = a(aa -aa)+a(aa -aa) + a(aa -aa) 11 22 33 32 23 12 31 23 21 33 13 21 32 31 22 Expansion of the determinants D , D , and D gives 1 2 3 D=y(aa -aa)+a(ya -ya) + a(ya -ya) 1 1 22 33 32 23 12 3 23 2 33 13 2 32 3 22 D =a (ya -ya )+y(a a -a a ) + a (a y -a y) . 2 11 2 33 3 23 1 31 23 21 33 13 21 3 31 2 D =a (a y -a y) + a (a y -a y)+y(a a -a a ) 3 11 22 3 32 2 12 31 2 21 3 1 21 32 31 22 For an example, consider the determination of an h value for y = 0.336 at the 0.001 probability level. From table 1 appropriate values are taken and placed into (12) to give the system of equations, 0.09a + 0.3a + a = 1.5009 1 2 3 0.16a + 0.4a + a = 1.4554 1 2 3 0.25a + 0.5a + a = 1.4142 1 2 3 The determinant of the coefficients is 0.09 0.3 'I D = 0.16 0.4 1 0.25 0.5 1 The remaining determinants are and 1.5009 0.3 1 D = l 1.4554 0.4 1 1.4142 0.5 1 0.09 1.5009 1 D = 2 0.16 0.25 1.4554 1 1.4142 1 0.09 0.3 1.5009 D = 0.16 0.4 1.4554 O 0.25 0.5 1.4142 17 Evaluating these determinants and applying Cramer's Rule gives a = 0.214999, a = -0.6055000, and a = 1.663200. 1 2 3 Substituting these values into (12) gives the required interpolating parabola, h(y) = 0.214999 y 2 - 0.6055000 y + 1.663200. Thus, for y = 0.336, h ■ 1.4840. 6. Calculation of the argument h/t and t Now that the interpolation of h values has been discussed, let us return to the calculation of h/t. As previously stated, this value may be obtained by taking the inverse of (11). P p (x) has a value given by (10). An initial estimate of h/t can be found using Hasting's (1955) rational polynomial approximation, c + c r + c r 2 x o = r " 1 ; a r + d r2 I d r3 + MQ) , (13) 1 2 3 where r = [ln(l/Q 2 )]\ Q = 1-P ni (x), and |e(Q)| < 4.5 x 10"\ The constants are c Q = 2.515517 d x = 1.432788 c = 0.802853 l d 2 = 0.189269 c = 0.010328 2 d 3 = 0.001308 Using x as a starting point, Newton's Approximation Method may be employed to obtain a better estimate of the true value of x. Thus, write f(x) = (x) - P m (x) and calculate x i = x i _ i - (f(x i _ 1 )/f'(x._ 1 )), i = l,...,m . (14) Let 0(x) denote the standard normal distribution integral, (x.) - (x. )] is less than some specified value. 18 The use of Newton's Approximation (14) requires the evaluation of the normal probability integral (15) for each successive approximation. This evaluation may be accomplished by using a power series or a polynomial approximation. Although quantiles presented in this paper were computed using a routine which employed the power series given below, a polynomial has also been used which can provide the same accuracy as the power series and effectively reduce computation time. The power series is 2n+i (x) = 0.5 + '(xj is the normal density function defined by (16). The following polynomial expression (Hastings, 1955) is used in the program presented in this report for computing normal probabilities in conjunction with Newton's Approximation: 4>(x) = 1 - «T(x) (f x c + f 2 K 2 + f 3 S 3 + f^ + f 5 C 5 ) + e(x) (18) where c = (1+qx) -1 and |e(x)| £7.5 x 10" 8 . The constants are q = 0.2316419 f = 0.319381530 f, = -1.821255978 1 h f =-0.356563782 f = 1.330274429 2 5 f 3 = 1.781477937 When x is returned from (14), it is the value h^t. The value h may be m obtained from the appropriate table or by interpolation as described pre- viously. Division of x by h and squaring gives the desired quantile t. As an example, compute the quantile for P Q (t) = 0.001 and y - 0.336. It was y determined earlier that for this probability and y» h was 1.4840. Thus, from (11) P (x) = (1 + ((.001) l/2(o * 336) ))/2 = 0.5000171666. Entering (13) with this value yields the first approximation x . Using Newton's Approxi- mation (14) yields for the third approximation, x 2 - Division of x 2 by h and squaring gives the desired quantile, t = 0.84077663789 x 10~ . This quantile produces a probability of 0.10000569989 x 10" 2 . While the quantile just obtained may be quite satisfactory for practical 19 reasons, it may be desirable to obtain an even better estimate of the speci- fied quantile. Let us then use the quantile obtained as an initial estimate and invoke the use of Newton's Approximation in connection with the gamma distribution function. Recalling (14), i.e., x i ■ Vi " ( f ( x i-i)/ f ^ x i-i)) V = If---.*) » (14) let f(x) = [F(t;y) - p a (t)] where P a (t) represents the probability for which we desire a quantile and F(t;y) denotes the computed probability for a quan- tile given the value of the shape parameter y. Now f (x.) = F(t;y) where F(t;y) is given by (6). The derivative of F(t;y) is then the gamma density function, i.e., F'(t;y) = (r( Y ))-i t^V* . (19) Recalling that the recurrence formula for the gamma function is r(y+l) = yr(y), then (19) may be rewritten as F'(t;y) = y(r(y+l))-i t^e"* . (20) The gamma function r(y+l) may be evaluated using an approximation by Hastings (1955): r(y+l) = 1 + b y + b 2 y 2 + b 3 y 3 + b y k + b 5 y 5 + b g y 6 + b y 7 + b 8 y 8 + e(y), (21) where <_ y <_ 1 and |e(y) | <_ 3xl0" 7 . The constants are b = -0.577191652 b c = -0.756704078 1 5 b = +0.988205891 b c = +0.482199394 2 6 b 3 = -0.897056937 b ? = -0.193527818 b, = +0.918206857 b = +0.035868343 k 8 When y>l, then the recurrence formula must be used to reduce the argument (y+1) to a value 1 < (y+1) <_ 2. The evaluation of (6) can be done using Pearson's Expansion (1957) which may be obtained by integrating (6) by parts. The expansion is F(t;y) = (r(y+l))-ltV t [l + t/(y+l) + t 2 /(y+l ) (y+2) + ...]. (22) Consider again the calculation of the quantile for a probability P (t) = 0.001 given y = 0.336. The value of the required quantile was 20 t = 0.840776663789 x 10" 9 which yields a probability 0.100000569989 x 10" 2 . If, however, this quantile is used as an initial estimate and used in Newton's Approximation, a better estimate is obtained. The new estimate becomes t = 0.840762399419 x 10" 9 which will yield a probability 0.999999999380 x 10' 3 . While variations in h in the third decimal place resulted in differences in computed quantiles in the second significant digit, application of Newton's Approximation produced quantiles which differed only in the sixth significant digit. In another instance, the output from Newton's Approximation yielded results which differed only in the ninth significant digit. For those combinations of P (t) and y for which [P (t)] 1 / 2y < 0.5 x 10" 10 , y y the inverse of expression (22) with some modifications was used to compute quantiles. To simplify expression (22), let t <_ 10" 8 and y < 0.1. The sum of the terms in brackets will yield (for practical purposes) a value of 1. Also, with e" «1 , expression' (22) becomes F(t; Y ) = (r(Y+l))" 1 t Y (23) Inverting (23), where F is estimated by P, yields t = e (Y)" 1 ln[(P)(r( Y+ i))] (24) The results obtained from this expression are compatible with those given by Wilk, Gnanadesikan, and Huyett (1962). 7. Comments The h values provided in this paper will yield quantiles which will give probabilities with a relative error of less than 0.05 percent for the selected probabilities with few exceptions. Interpolated h values also yield similar relative errors. For probabilities other than those provided, h values may be estimated from the tables and used in computing an initial estimate of the desired quantile. Newton's Approximation then may be applied to obtain the required accuracy. 21 VII. PROCEDURES FOR LARGE SHAPE PARAMETERS This section might appropriately be entitled "A New Approach to Quantile Determination." It documents a new technique for the computation of quan- tiles for the gamma distribution when the shape parameters are large in value, in particular, quantiles for 3 <_ y <_ 104 and 45 probabilities. The determination of quantiles for the gamma distribution has presented numerous difficulties. Most of these have resulted from the inability to invert the gamma distribution function in a direct manner. A technique was initially developed in section VI which permits quantile computation when the shape parameter is small, i.e., y < 1. This permits a significant reduction in computer processing time for programs requiring quantile computation. This technique was later extended to shape parameters less than 4 and is included in this program. The following technique follows in part from a procedure given by Wilk et al . (1962), but the quantile determination presented here is done in an entirely different manner. The same procedure used by Wilk et al . (1962) has also been used by Mooley (1973). We desire to determine x in the following expression given the shape parameter y. (Note that the standardized form of the gamma distribution function is used to simplify computations.) ^-mj t^'V* dt (25) Let xv = t, then, dt = rdv. Substituting in (25) gives (l 1 P(t) = TWJ T Y-i v Y-i e -TV Tdv , (26) where v is chosen at the upper limit such that t = t, i.e., v = 1. One of the properties of definite integrals is that if g(x) <_ h(x) within 22 J J the interval c <_ x < d, then / g(x)dx <_ /° h(x)dx. Let us apply this property to (26) to arrive at an expression which will give, upon integration, a means of estimating the lower limit of the desired quantile. By the given property, if (v Y ~ e~ TV ) <_ (v Y_1 ) where <_ v <_ 1, then we have (writing the complete expression 26) P(t) = 1 rW7 ri Y v-l t'V 1 -TV dv 1 rTTT t'V 1 dv (27) or P(t) Ir^nj • (28) Solving (28) for x gives a lower limit then for the quantile corresponding to the probability P(x): x = [P(t) • r( Y+ l)] 1/Y . (29) This expression is best evaluated using the logarithm of the gamma function since r(y+l) becomes extremely large as y increases. The logarithm of (29), with r(y+l) = Y r (y)> gives In x = - [In P(t-) + In r( Y ) + In y] • (30) After substitution of appropriate quantities in the right side of (30), subsequent exponentiation gives the desired lower limit of the specified quantile, x = exp [ - [In P(x) + In r(y) + In y]] • (31) The computation of In r(y) and P(x) will be covered later on. The computa- tion of P(x) is required in using Newton's Approximation when checking the convergence to the desired accuracy in the quantiles. It was in the course of working with the lower limit that the notion was conceived that a definite relationship might exist between the lower limit and the "true" or "exact" quantile. Indeed, comparison between known quantiles and lower limits indicated that a functional relationship existed, While a functional relationship has not yet been determined, an algorithm has been found which yields the ratio between the true quantile and the 23 lower limit (Jalickee et al., 1975). With the ratio for a selected proba- bility and shape parameter, the true quantile may be easily determined by multiplying the computed lower limit by the ratio. If the desired accuracy is not obtained, this quantile may be subsequently used in conjunction with Newton's Approximation to provide a better estimate of the true quantile. These ratios have been called "gamma quantile ratios." The input data for the development of the algorithm consisted of 4,725 ratios generated by a program using a Hewlett-Packard HP9830 computer. Some overlap occurred in the y = 20 - 22 range to provide continuity for both y-probability fields for which the algorithm was desired. Program execution began with a ratio estimate for the lowest y-probability combina- tion for the field being desired, i.e., y = 3 or 20 and probability = 0.001. Once the program produced the quantile to the desired accuracy, the ratio was computed, i.e., the ratio between the true quantile and the lower limit. This ratio was not only stored in an array but also used to generate the next desired quantile. Computations of this nature continued until the array was filled. These data were stored on tape for reformatting and retrieval at a later date. The algorithm developed uses 975 values counting the overlap mentioned above. For the field defined by y = 3 - 22 and probability = 0.001 - 0.999, there are two arrays (5 x 45 and 5 x 20); for y = 20 - 104 and probability = 0.001 - 0.999, the arrays become 5 x 45 and 5 x 85. The algorithm is of the form, f(x,y) = xy +xy +xy +xy +xy , (32) 11 2 2 33 h k 55 where the subscripts denote the column from which the x and y are taken. Going beyond five terms showed no significant improvement in the computed value. The algorithm input values are given in tables 4.1 and 4.2 of appendix 1. An example of the use of the algorithm is provided in appendix 2, If it is known that quantiles computed do not have the desired level of accuracy, it becomes necessary to employ an approximation technique which hopefully does not consume too much computer time. The method employed in previous programs, and found to be applicable here with slight modifications, is Newton's Approximation Technique. Convergence using this approximation 24 is quadratic, a most favorable characteristic. Quadratic convergence indi- cates that each error is roughly proportional to the square of the error in the previous approximation. This means that the number of correct digits nearly doubles with each iteration. This popular algorithm takes the form, f(x n-i ) x n = x n-i ■ Hx^p ' (33) J.L. where x denotes the n approximation to a root of f(x) = 0. In the quotient term, the f(x ) denotes the function for which the root is desired but evaluated using the (n-1) approximation; the prime in the denominator denotes the first derivative of the function evaluated using + h the (n-1) approximation. Iterations are continued until the required degree of accuracy is obtained. To accommodate extremely large numbers which were observed to halt program execution during the course of quantile extraction, a slight change in (33) was required yielding x n = x n-i " ex PHnf(x n _ 1 ) - Inf (x n-1 )] . (34) Now let us consider the terms in (34) which are in brackets. Define f(* n _ 2 ) as f(x n _ 2 ) = F(x n _ 1 ) - P (35) and the derivative becomes f ' * F '<*n-i> < 36 > since P is a constant, i.e., a preselected probability value. In general, F(x) is the gamma distribution probability function so that the first derivative F'(x) is the probability density function. In effect we have, within the brackets of (34), the logarithm of the difference between the probability of the quantile approximation and the desired probability P for which the quantile is required less the logarithm of the gamma density function. The exponentiation of the term in brackets gives a correction to the (n-1) approximation. The one problem which arises in using the form presented in (34) is that concerning logarithms of negative quantities. This can, however, be circumvented by using absolute values of the differences in probabilities and adding (subtracting) the corrective term when there is an underestimate (overestimate). 25 The evaluation of the probability F(x) required in (35) is accomplished using Pearson's Series which may be derived easily by integration of (25) by parts. Although the convergence of the series is slow, it has been used extensively. A technique is available for the evaluation of the incomplete gamma function when the shape parameter y >_ 50 (Fisher, 1973). The accuracy of this algorithm increases with an increase in shape parameter with a subsequent decrease in the number of terms in the approximation. In essence, the incomplete gamma function is approximated by a finite series expansion -- an Edgeworth series. Overflow and rounding errors are avoided to a great extent using this method. The execution time of this algorithm is virtually independent of y and the quantile (upper limit). Other algorithms of the type of Bhattacharjee (1970) have an execution time which, for a fixed y value, is a bell-shaped function of the upper limit of integration. At some later time, this algorithm could be inserted into existing programs if savings in program execution times could be seen. The form of Pearson's Series which can be used is given in its full form below. For the original form, reference may be made to several papers (Pearson, 1957; Thorn, 1968). The probability F(t) is given by F(t) = exp(yln T-T-lnr(y)-lnY) oo n 1 + z exp(nln x-zln(Y+m)) n=i m=i (37) In (35), when F(t)-P < 10" 9 , computation Computation of this series may be terminated when, for any n, n exp(nln x-zln (y+m)) £ 10 m=i using Newton's Method may be halted. The density function required in (33) is used in the logarithmic form in (34), i.e., In f(x) = -t + (y-1) • Inx - lnr(y) . (38) There remains now the evaluation of In r(y) which appears in several places. Once the value is determined, it may be stored for further use. Although the expression is used in its nested form for the computer evaluation, the logarithmic form of Stirling's Series as used by Mooley (1974) is given here by 26 A- A A, 3 In r( Y ) « (y - 0.5) In Y - y + 0.5 ln(27r) + ^ - j^? + 30 y 5 A A A A A 4.5 6.7 8 (3g) " 56y 7 90y 9 " 132 Y n 182y 13 " 240 Y 15 ' where the Bernoulli numbers k 1 through A 8 are A 2 = 1/6, A 2 = 1/30, A 3 = 1/42, A^ = 1/30, A 5 = 5/66, A 6 = 691/2730, A 7 = 7/6, and A 8 = 3617/510 Mooley (1974) reports that, for y >. 4, an accuracy of 11 decimal places can be obtained. To summarize, the following are the steps in determining a quantile: 1. Compute the lower limit of the quantile. 2. Use the ratio algorithm to compute the quantile ratio. 3. Multiply the lower limit by the ratio to obtain the desired quantile. 4. If the desired quantile is for other than one of the integral gamma values or the selected probability values, use Newton's Approximation to determine the quantile. In this case, use the ratio for the gamma value and probability on the lower side of the tabled combinations. 27 VIII. MIXED DISTRIBUTIONS Some data sets form a mixed set of distributions. The simplest mixed set consists of two subsets of data: 1. All data equal to or less than a, the origin. 2. All data greater than a. Where the origin a is zero, the mixed set consists of 1 . The subset of zeros and 2. The subset of measured quantities. Thus, after Thorn (1951), H(x) = q + p G(x) , (40) where q is the zero-set empirical probability, p is the measured-set probability, and G(x) is the gamma distribution function for the measured set. For example if q = 0.40 and p = 0.60, 40% of the observed values are zero and 60% of the observed values are greater than zero. Then, the cumulative probabilities of amounts greater than zero develop from the solution of G(x). These probabilities then are multiplied by 0.60 and added cumulatively to the initial 0.40 probability for the zero. If a is not a zero, then the q = 0.40 would apply to values <_ a. Also, p refers to values > a. The above procedure, utilizing the simplest mixed distribution, is part of the present computer program. Neither the model nor the program considers or allows for mixtures within the set of measurable quantities. 28 IX. GAMMA DISTRIBUTION FUNCTION COMPUTER PROGRAM Elderton (1953) provides the moment estimate procedures for the origin parameter a as indicated previously. Thorn (1958, 1968) provides the requisite information and equations to provide the maximum likelihood (ML) * * and Thorn estimates of the scale and shape parameters 3 and y. The computer program given in appendix 3 initially follows after Bark and Hofman (1960). Since that time and through much usage, discussions, and changes, resemblance to the original program decreases. The previous program (Crutcher et al . , 1973) may provide inadequate approximation for values of the probabilities when the shape parameter y is less than 0.50. In this region, the asymptotic portion of the gamma function distribution, the slope of the curve, is almost indeterminant. Small changes in the shape parameter cause extreme changes in the function. Pearson (1922) discusses this problem. Where computers of extremely large capacity are available, the approximations may succeed at low gamma and low probability values, though numbers as small as 10~ 35 are reached before failure. When dealing with real data, such low gammas and low probabilities are not of too great importance. However, in terms of reliability problems, these may be important. Therefore, the present paper presents work done on this problem in the development of approximation algorithms or techniques. Caution is still needed when using this program for shape parameter values < 0.10. The FORTRAN IV computer program that forms appendix 3 employs the Univac Series 70/45 computer. Use with any other computer may require a few changes, but these will be minimal. Other options may be inserted, and changes may be made by the user to satisfy his particular requirements. Figure 2 illustrates in tabular output form the application of the gamma model to the weekly rainfall distribution at Auburn, Alabama. The first week of the climatological year, March 1-7, for 36 years with measured precipitation amounts in 40 of the years constitutes the data set. Figure 3 depicts the statistics for the second week of the climatological year, March 8-14, for 35 measured precipitation amounts in 40 years. 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K z DL < ooooooo-h— coo>o>o-HcM (Mffi^-iAh-ajOH-tooinA Q- o 3Z 3 UJ 1T> Ul a of a: rg tt toS C\i a. co oooooorjifia)r4 -j-r-o>c\jtrico^^'J-i v -o>cMmco ALABAM X ui or D. oooooooOHHHHNNNmnmffl-t't^iniftflifto o jj o i — f~r-—cocococooJ*o>o , > ooooooooooooooooooooooooooo ooooooooooooo o ■> If) z -t z ' z ■a i ooooooooooo»o»o>o>i>0'cj>cocococoi v -r-r^i v -i^>o ■o^iflininininn^rfoh* AC *t I- t- ooooo^mmir\r-<-r-o«MOO-»i«\tf>o-3-r-ooivj-4 , ir\ iniDOmmioor^cy-J/ir* 3 CB •a - oo^o^'tinrimHoin^aO't^in'OHininoooOH •tiftW<0<-if^a)'-»o»(r»OiHiX I CM~ r-j CM l- 1- z < HOHOIM«H(llH^«sri < MV^mroom«^hh«OOh nnj'ONi>--tioO'«woir\H OCOOC»-iOOOO'tf-0--'OOO^rsJOO^-iO^^OO^ r-ic^r-i;^-iorctoc«^->$' i— —1 in <" CO D u D) _o O O E "u a) a> c O u a) d) D> c i_ _Q D < o * I £ o .5? co -° T o »- CO Q. .2 fe 2 ^ ll 0) a. >s CO a> D) 31 maximum of 52. Fifty-two is also the maximum data set input. This latter restriction, of course, may be bypassed if the option starting with known estimates of the scale and shape parameters is used. Parts A and B in figures 2 and 3 and in data output divide the tabulations into two sets of columns. Part A, shown on the first line, provides the following: 1 Station Identification 2 I Sample number 3 J Number of duration periods in sample 4 NX Number of data excluding zeros 5 NNX Number of data including zeros 6 XBAR Arithmetic average of data excluding zeros 7 ALPHA Origin value 8 BETA Scale parameter estimate, -BETA STAR 9 GAMMA Shape parameter estimate, GAMMA STAR 10 X2 X 2 for chi-square test 11 PROB Probability of a chi-square equal to X 2 above 12 K-S The largest difference in probability between the theoretical and empirical distribution curves. This is the Kolmogorov-Smirnov test statistic (Smirnov, 1948). Part B comprises 13 columns of output information that provide the following: 1 Sequential guidance 2 Data in order of observation or record. These are x or y or transforms of y such as (y-a) or (y-a)/e. 3 Ordered data of column 2 4 Ordered data of column 2 divided by the scale parameter 3 of column B-2 data. If the transform ((y-a)/3) is used, columns B-2 and B-4 ought to be identical except for rounding error. 5 Empirical probability of the ordered data. The expression (n-c)/(n-c+l) provides the probabilities where n is equivalent to NX of part A and c = 0.44 (Gringorten, 1963). NX is the number of nonzero data. A program option permits a change in the value of c. 6 Variate quantile associated with the empirical probability of column 5 with the scale parameter 6 set equal to unity. 32 7 Variate quantile associated with the empirical probability of column 5 with the sample scale parameter 3 (beta star) shown in part A. 8 Fifty- two or less arbitrarily selected cumulative theoretical probability values for which columns B-9 and B-10 respectively show corresponding cumulative quantiles and amounts. A program option permits change in these, but allows for no more than 52. 9 Cumulative quantile values of the distribution corresponding respectively to the cumulative probability values of column B-8. 10 Cumulative values of the distribution corresponding respectively to the cumulative probability values of column B-8. Multipli- cation of values in column B-9 by the sample 6 (beta star) value of part A provides column B-10 data. 11 Consider the base. The base is only the distribution of nonzero amounts shown in columns B-2 and B-3. The number of data is the NX of column A-4. Column B-ll then gives the probabilities of occurrence of amounts equal to or less than selected nonzero amounts shown in column B-12. This column is labeled "GRAPH" to indicate that this may be used to graph the set of nonzero amounts. 12 Arbitrarily selected cumulative amounts. The maximum number of amounts is 52. A program option permits change to less than 52 amounts. The option also provides for the amounts to be scaled in terms of the mean of the nonzero amounts. 13 Probabilities of exceeding the arbitrarily selected cumulative amounts shown in column B-12. This is the mixed distribution. If NX = NNX in part A, columns A-4 and A-5 (i.e., if the original distri- bution has no zero amounts), then column B-13 is the complement of column B-ll. The plot of columns B-ll, B-12, and B-13 (as one set) and columns B-8 and B-9 (as another set) should plot on the straight line of the graphs shown in this report. The data of column B-12 should be scaled by division by the scale parameter 3. The empirical probabilities and empirical amounts shown 33 in columns B-5 and B-7 plotted on the graph will show visually and sub- jectively how good the line of best fit fits the data. Wherever the approximation routines fail for a particular quantity or probability level, this will be noted in the output. Usually, enough levels will be available so that the loss of a level or two is not important (i.e., interpolation will suffice). If too many levels are noted, then the program routines generally will be inadequate because of difficulties previously mentioned in the asymptotic portion of the distribution. For most purposes (in the analytical sense), if the gamma model is accepted without question, columns B-8 and B-10 or columns B-12 and B-13 provide the desired information based on the data sample. One set is the inverse of the other, though different levels may be and generally are used. 34 X. GRAPHS Graphs used here are based on the work of Crutcher, Barger, and McKay (1973). The design of figures 2 and 3 permits the easy extraction of infor- mation for manual or automatic plotting. The program prior to subroutine plot (line 1310) in appendix 3 incorporates the command instructions for the CALCOMP Plotter System to a drum computer and/or Computer Output to Microfilm (COM). An option permits selection of any or all the graphs and either or both plotters. These particular instructions can be used only if the CALCOMP computing package is available. However, if this is not available, modifi- cations based on details given here can be used to control any x-y plotter. Four graphs are developed here. Many others could be developed from the great amount of information contained in the computer output exemplified in figures 2 and 3. These graphs developed here permit: (1) a visual assessment of the theoretical density based on the parameter estimates, (2) a visual assessment of the chi-square statistics by means of a histogram, (3) a visual assessment of the Kolmogorov-Smirnov statistics by the use of a cumulative ordered interval data overplot on the theoretical cumulative intervals; and (4) a visual assessment of the line of best fit which has a line of slope 1, the diagonal line on the graph. An ordered data overplot is provided. The preparation of the graphs are now discussed in the above order. 1 . Density Curves Figures 4a and 5a provide a truncated density curve for the non-zero data in figures 2 and 3, respectively. The size of the plot on microfilm is as large as the Computer Output to Microfilm (COM) permits, but the plot can be enlarged photographically to any desired size. The vertical scale maximum is standardized to 1 and divided into fourths. The maximum value is always located on the horizontal scale at a quantile computed for a shape parameter 35 >0 (b) FIGURE 4a. AUBUBN, ALABAMA PRECIPITATION PROBABILITIES MARCH 1-7, 1930-1969. 0.10 0.20 0.30 0.40 0.50 0.60 0.70 PROBABILITY FIGURE 4b. AUBURN, ALABAMA PRECIPITATION PROBABILITIES MARCH 1-7, 1930-1969. 0.70 0.80 0.! PROBABILITY FIGURE 4c. AUBURN, ALABAMA PRECIPITATION PROBABILITIES MARCH 1-7, 1930-1969. .950 II I .010 .250 .500 .750 .100 .350 .650 .650 PROBABILITY FIGURE 4d. AUBURN ALABAMA PRECIPITATION PROBABILITIES MARCH 1-7, 1930-1969. 36 (FMAX >0 1 0.629 STN I J N* NNX XBRR BETA CflA I04gg 2 1 35 40 L-gtS 1 -003 1- 7- >0 - ! -7 5- -5 > z 3 4- o E -A 3 - -3 2- -2 1 - -1 FIGURE 5«. AUBURN. ALABAMA PRECIPITATION PROBABILITIES MARCH 8-14, 1930-1969. 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.9 PROBABILITY FIGURE 5b. AUBURN, ALABAMA PRECIPITATION PROBABILITIES MARCH 8-14. 1930-1969. 0.70 0.80 O! PROBABILITY FIGURE 5c. AUBURN. ALABAMA PRECIPITATION PROBABILITIES MARCH 8-14, 1930-1969. n i i i r .010 .250 .500 .750 .950 .100 .350 .650 .850 PROBABILITY FIGURE 5d. AUBURN. ALABAMA PRECIPITATION PROBABILITIES MARCH 8-14. 1930-1969. 37 value one less than the actual shape parameter estimate. That is, the modal value of a gamma distribution is always at (y-1). When the shape parameter is 1 (the exponential distribution), the modal value is at 0. When the shape parameter is less than 1, the modal value is indeterminate. In the last case, the modal value will be truncated at the top of the square. The legend below the figures contains the modal maximum. If it is equal to or greater than 1000, computing will be stopped and the legend will contain the statement that the modal maximum is equal to or greater than 1000. Equation (41) is the analytic expression of the density function f in terms of the quantile t and the shape parameter, y f = f(t, Y ) = (r(Y))" 1 t Y "i e"* (41) In the calculation of the X 2 statistic, an option permits the selection of any number of equal probability class intervals. If the option is not exercised, the default option is ten. It is impossible to compute the plotting quantile for 100 percent. As the number of intervals is optional, it seems best here to simply drop off the last interval, that is, to truncate the distribution at the (k-1) interval. If the user is not satisfied with this, the program may be modified to permit extension to any predetermined probability level (except 1.00) or quantile. In most cases, a representative density curve will be obtained. Comparison of this density curve with the curve expressed for a given shape parameter (see figure 1) will provide a consistency check. 2. Histograms Figures 4b and 5b, respectively, are truncated histograms for nonzero data contained in figures 2 and 3. Truncation is made here for the same reasons given above. That is, the last probability interval is not shown. This does not invalidate comparisons but some care must be exercised. The expected frequency histogram is shown by the dashed line. Simply, it is a value equal to NNX divided by the number of intervals k, in this case to ten. Where 20 cases are used, the expected frequency in each equal proba- bility class interval is 2.0. Where there are 25 cases, the expected frequency shown would be 2.5. 38 ft The actual frequency histogram is shown by the solid lines. It is the differences of these intervals, the expected frequencies less the observed frequencies, that are used in the calculation of the X 2 statistics (X not chi). The frequency difference is squared and then divided by the expected frequency. One such value is available for each equal probability class interval. It is the addition of all of these values that gives the value of X 2 which is shown on line A of figures 2 and 3. The horizontal scale is plotted in terms of quantiles which are linear for the probability levels dictated by the option of class intervals. For example, 10 class intervals would select probabilities of 10, 20, 90 percent, while 20 class intervals would select probabilities of 05, 10, 95 percent. There- fore, the distances between the vertical scale of the frequency (bar) histogram will be variable and will depend on the shape parameter. The narrowest bar will be in the neighborhood of the modal point of the density curve. These figures are plotted to permit an assessment of symmetry. If the figures imply considerable asymmetry, the use of the X 2 statistics or the chi-square test should be suspect. To begin with, the chi-square test is not a powerful test. The greater the number of data in each interval, the better the test. However, an increase of the number of class intervals does not necessarily increase the power of the test [Tate and Hyer (1973)]. For this test, a maximum of 20 equi-probability class intervals is suggested. These histograms are quite different in appearance than those where equal class intervals in terms of the quantities or in terms of the variate are chosen. Symmetry will be indicated when there is a single horizontal line across the figure at the expected frequency. That is, the solid horizontal line will be over the dashed horizontal line. Asymmetry will be implied when the solid lines are low on the left and high on the right or high on the left and low on the right. These figures must not be used for assessment of areas under the curve. The areas within each interval under the density curve would be equal. There- fore 3 it is only the vertical differences between the dashed lines and the interval tops that are to be used here. 39 These tests apply only to the non-zero data, i.e., the data greater than the origin. 3. Cumulative Frequency Diagrams The cumulative diagrams, figures 4c and 5c from figures 2 and 3, respec- tively, permit assessment of the Kolmogorov-Smirnov statistics as discussed by Crutcher et al . (1973), Kolmogorov (1933), and Smirnov (1936, 1948)., The horizontal scale is linear in terms of quantiles as in figures 4 through 6 and again is marked in time of equal -probability class intervals. The vertical scale is in terms of frequencies with a maximum value equal to the number of nonzero data, NNX of figures 2 and 3, respectively. Dashed lines provide the expected frequency cumulative diagram, while the solid lines provide the observed frequency cumulative diagram. Where they are the same, only the solid line will show. Differences are indicated by the vertical or horizontal hachuring. When the expected frequency is greater than the ob- served frequency, the hachuring is vertical. The greatest difference is indicated by both vertical and horizontal hachuring. Where there are ties, only one will be indicated. The greatest difference between the cumulative curves is the basis of the Kolmogorov-Smirnov (K-S) test discussed by Crutcher et al . (1973). Though there appear to be discrete data, they are from continuous distributions. Figures 2 and 3 present the K-S statistics as the last item in line A. These tests apply only to the nonzero data. Table 5 (appendix 1) provides data to test the significance of the K-S statistics [Lilliefors (1967, 1969, 1973) and Crutcher (1975)]. For example the values of the K-S statistics of figures 2 and 3, 0.061 and 0.057 for n values of 36 and 35 with gamma values between one and two, are not signifi- cant. Therefore, the null hypothesis that the gamma fit is not significantly different from the data set is not rejected. The gamma fit is then used. 4. Ordered Data Model Fit This type of graph as presented in figures 4d and 5d from figures 2 and 3, respectively, is perhaps the most important of the four presented here. The scaling is the same in all of figures 4 and 5 in that it is linear in the quantiles. The vertical scaling is in terms of quantiles on the left and 40 quantities on the right. Ordinarily, the units are obtained by multiplying the quantiles by the scale parameter; but in order to scale properly for plotting, division is required. The vertical scaling on the right, marked quantities, is then in units of the data, in this case inches. The type of data, i.e., precipitation, is not placed on the scale. The scaling numbers on the right will be integer values only when the scale parameter is an integer. Plotting, as always, is in terms of the linear quantiles marked in terms of probabilities along the abscissa, and in the ordinate quantiles on the left and data units on the right vertically. The line of best fit is the 45 degree line, line of slope 1, running diagonally from the lower left to the upper right of the square. The symbols represent the relative position of the actual observed data. A visual assessment of the fit is easily made. Here, the fit to each data set appears acceptable. This decision also is supported by the non-rejection of the x 2 and the Kolmogorov-Smirnov tests of the null hypothesis. There are two probability scales on the horizontal axis. The tick marks extending upward on the inside refer to the probabilities for amounts greater than zero shown in columns 4 and 12 of figures 2 and 3. The tick marks ex- tending downward refer to the probabilities for amounts equal to or greater than zero, i.e., the zeros are included. The reference columns are 8 and 10. Figure 6 is presented as a modification of figure 4d to illustrate the relationship between figure 2 and figure 4d. Columns 12 and 11 are first chosen. The quantity level of 6 inches of precipitation and its corresponding probability of 0.981 are chosen. The quantity 6 inches is divided by the scaling factor 1.202 to obtain 4.991. The first step on the graph is to draw a horizontal line parallel to the baseline at 4.991 (on the left as a quan- tile) across figure 4d as in figure 6. In the second step, in units of the variable, the value of 6.00 will be read on the right. Third, extend a line vertically downward from the intersection of the diagonal line of best fit and the line drawn in step 1. Mark this with an upward extending tick from the baseline and label with the appropriate theoretical probability 0.981. This may be repeated for each of the paired values of columns 12 and 11. 41 II I .010 .250 .500 .750 .100 .350 .650 .850 PROBABILITY FIGURE 6. AUBURN, ALABAMA PRECIPITATION PROBABILITIES MARCH 1-7, 1930-1969. 42 As plotting from columns 4, 6, and 5 of figure 2 is also for amounts greater than zero, these are now plotted and compared on the same scale indicated by the upward extending tick marks inside the square. For the last point plotted on the graph, the appropriate values are 4.418, 5. 189, and 0.984, respectively. As a fourth step, the quantile value of 4.418 on the vertical is plotted against the empirical quantile value of 5.189 and marked with some symbol, here a small cross (/). Through the cross, extend a line vertically downward as step 5 and mark with the appropriate value of 0.984 if required. As step 6, extend the vertical line in step 5 upward to the diagonal line of best fit and then horizontally to the units scale on the right where the appropriate quantity of 6.237 is then read or approximated. To keep the numbers from over-printing too frequently, only the selected probabilities above 0.50 are plotted and marked although all selected proba- bility tick marks are shown. If the user should sometimes use more proba- bility levels, over-printing will have to be accepted or selection for printing changed in the program. Even then, when the shape parameter is small, there may be difficulty which could be avoided by change of scale or curtailment of printing. For amounts associated with selected probability levels, the data in columns 8, 9 and 10 of figure 2 are used. At selected probability levels the appropriate amounts or less are given. For step 7, the probability level of 0.950, the quantile 3.765, and the unit value of 4.525 inches are used. A line is extended horizontally from 3.765 on the left across the diagonal line of best fit to the right-hand side where the value of 4.525 is read. As step 8, a line is extended downward to the base, marked with a tick mark extending downward and labeled as 0.950, the corresponding probability. Repeating these steps for other paired values in columns 8 and 9 will provide an appropriate probability scale. A full grid, not shown here, can be ob- tained by change of instructions to draw the entire lines rather than just the tick marks shown here. 43 XI. GRAPH PAPER Many investigators determine a line of best fit of many models to a set of data. Then they choose the model which appears to provide the best of the many lines of best fit. This may be done in terms of a visual fit or in terms of a least squares of deviations fit. This is a dangerous policy for there are errors of the first and second kind involved. No causal relation- ships can be drawn in such a post analysis. If there is an "a priori" design of an experiment and a variable(s) is (are) under the control of the experi- menter, then some causal or physical relationships might be established. In all of the foregoing it has been assumed that the gamma distribution is an acceptable model. As pointed out by Crutcher, Barger, and McKay (1973), rainfall may be likened to a study of reliability or failure. They provide graph paper covering several ranges of tables provided by others such as Thorn (1968), Harter (1964), Wilk, Gnanadesikan and Huyett (1962), and Pearson (1922). Tables are used because (except for the exponential distri- bution when the shape parameter y equals one) when G(y) = 1 - exp (-y), the inverse form G~ cannot be put in closed form. Some tables are provided in appendix 1 . At this point it is relevant to point out that, in using moment estimates, the following relationships hold. As a rough cross-check, the arithmetic mean and the sample standard deviation can be used to check the graphical estimates. The mean is equal to the product of the scale and shape parameters The variance is equal to the shape parameter times the square of the scale parameter or is equal to the mean times the scale parameter. This allows one to quickly calculate two of the parameters when the other two are avail- able. In symbolic form these relationships are x = 3 y or 3 = s 2 /x = a 2 /y S 2 = 3 2 Y Y = X 2 /s 2 = {l 2 /a 2 (given g and y) (given x and s 2 as y and a 2 ) In this sense the shape parameter is equal to the reciprocal of the square of the coefficient variation. 44 Crutcher et al . (1973) present graph paper for the gamma distribution covering several ranges of the shape parameter. This paper also may be used to estimate the scale and shape parameters. Figure 7, modified from Kao (1968), illustrates another form of graph paper for plotting and graphical estimation of parameters. (Kao's paper was not available to the authors prior to the preparation of the 1973 paper.) The hypothetical data and procedures used by Kao and given in appendix 1 as table 6 are used here to illustrate further the application of the gamma distribution to reliability problems. Gupta and Groll (1961) provide further discussion. Gringorten (1963) and Crutcher et al . (1973) use (i-0.44)/(n+0.12) rather than i/(n+l) to determine plotting positions. Blom (1958) and Kimball (1960) discuss this problem in detail. The gamma probability paper (figure 7) has two ordi nates. The right-hand one is the y-scale which is linear and the left-hand one is the p-scale which is equal to G(y) for y = 1 (exponential case). The extreme left portion of the paper gives the p-scale for other y-values in the range of (0.5 to 5.0). In plotting the data, one chooses the p-scale by trial and error (there are infinitely many in the range) until the plot appears to be "linear." The diagram at the upper left corner reminds us of the estimates for a, the location parameter, and 3> the scale parameter. Life quality characteristics such as mean life, standard deviation of life, and median life can be esti- mated graphically through the estimates for a and 3. For each estimate for y (the y value which yields a linear plot), there is a corresponding estimate value for u/3 and a/3. The additional scales on the top of the graph paper are for small values of y- Knowing y/3 and a/3 and estimates for a and 3, y and a can be estimated. Other life quality characteristics of interest are reliability functions at some x, and reliable life at any specified reliability index r. These quantities will now be defined and estimated graphically. The reliability function R(x) is defined as 1 - F(x) for any x. Since the probability plot is in itself an estimate of F(x), the estimate of R(x) can be readily obtained by reading the [1 - F(x)] value from the appro- priate p-scale for any x value. The reliable life p„ is the inverse function ,-i r of R(x), or p = R~ (r) which is solved on the probability plot by starting 45 CM CV- \D _ -* 8^ CO £- sO UA . 6 O &> s Q\ O O 00 -Sh, 3 R R -* R. ON . • 01 01 || c\ H 00 O S- +j f0 en i s- ta o o "a. CD cu s- 3 en >|cq. a] CO. 46 from (1 - r) value at the appropriate p-scale horizontally to the probability plot and reading p from the abscissa. The median life is equal to p 50 — a special case of the reliable life. Figure 7 shows that three trials on the data have been made, one for each of the following y values; y = 1> y = 3, and y = 5. Apparently, y = 3 yielded the most linear plot, with y/3 = 3 and a/e = 1.73. Hence, the estimates for the gamma parameters are Location: a = 2 khr. Scale: 3 = 2.8 - 2 = 0.8 khr. Shape: y = 3 (no dimension). A few further calculations give the following estimates: Mean = a + $(y/e) = 2 + 0.8(3) = 4.4 khr. Standard deviation = 0(a/e) = 0.8(1.73) = 1.38 khr. Reliability function at 2.3 khr. = 0.99. Reliable life at 90% = 2.9 khr. Reliable life at 95% =2.6 khr. Median life = 4.2 khr. 47 XII. FUTURE MODIFICATIONS TO THE PROGRAM The following are six expected modifications planned for the computer program either as a part of the program or as separate programs: 1. A subroutine for the determination of a better theoretically and practically acceptable location (origin) parameter estimate. 2. A possible subroutine for the debiasing of the maximum likelihood and Thorn (1958) shape and scale estimators. 3. Separate gamma distribution programs for one data string to provide (a) quantiles for specified probability levels and (b) probabilities for specified quantiles. 4. Separate normal distribution program for one data string for cases where the shape parameter estimate is greater than 100 to provide (a) quantiles for specified probability levels and (b) probability levels for specified quantiles. 5. The development of confidence levels (tolerance bands) for the lines of best fit. The binomial distribution has been used. This will be compared with Monte Carlo techniques. 6. The examination of possible better estimators for the parameters of the gamma distribution, particularly where all three parameters origin, scale, and shape, are unknown. 48 REFERENCES Anderson, C. W., and Ray, W. D., 1975: Improved maximum likelihood estimators for the gamma distribution. Communications in Statistics , 4 (5), 437-448. Andrews, Fred C. (University of Oregon, Eugene), and Barger, Gerald L. (Laboratory for Environmental Data Research Environmental Data Service, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, DC), 1956 (personal communication). Barger, Gerald L., 1964: The NWRC at Asheville can help you! Proceedings of the Institute of Environmental Sciences , Philadelphia, PA, 10 pp. Barger, Gerald L., Shaw, Robert H., and Dale, Robert F., 1959: Gamma Distribution Parameters From 2- and 3-Week Precipitation Totals in the North Central Region of the U.S. Agricultural and Home Economics Experiment Station, Iowa State University, Ames, IA, 183 pp. Barger, Gerald L., and Thorn, Herbert C. S. , 1949: Evaluation of drought hazard. Agronomy Journal , 41 (11), 519-526. Bark, L. Dean, and Hofman, Larry B., 1960: FORTRAN II program determining precipitation probabilities from a fitted gamma distribution. Contribution No. 94, U.S. Weather Bureau Contract Cwb 10257, Department of Physics, Kansas Agricultural Experiment Station, Kansas State Univer- sity, Manhattan, KS, 10 pp. Bhattacharjee, G. P., 1970: Algorithm AS 32; the incomplete gamma integral. Applied Statistics , 19 (3), 285-287. Birnbaum, Z. W. , and Saunders, S. C, 1958: A statistical model for life lengths of materials. Journal of the American Statistical Association , 53 (281), 153-160. Blischke, Wallace R. , 1971: Further results on estimation of the parameters of the Pearson Type III distribution. Aerospace Research Laboratories ARL 71-0063, Contract No. F33615-70-C-1136, Project No. 7071, Wright- Patterson Air Force Base, OH, 48 pp. Blischke, Wallace R., 1974: On nonregular estimation: II. Estimation of the location parameter of the gamma and Weibull distributions. Communications in Statistics , 3 (12), 1109-1130. Blom, Gunnar, 1958: Statistical Estimates and Transformed Beta-Variables . John Wiley & Sons, Inc., New York, NY, 176 pp. 49 Campbell, G. A., 1923: Probability curves showing Poisson's experimental summation. Bell System Technical Journal , 2 (1), American Telephone & Telegraph, New York, NY, 95-113. Chapman, Douglas G., 1956: Estimating the parameters of a truncated gamma distribution. Annals of Mathematical Statistics , 27, 498-506. Cohen, A. Clifford, Helm, F. Russell, and Sugg, Merritt, 1969: Tables of areas of the standardized Pearson Type III density function. NASA Contractor Report CR-61266, George C. Marshall Space Flight Center, Huntsville, AL, 8 pp. plus tables. Crutcher, Harold L., Barger, Gerald L., McKay, Grady F., 1973: A note on a gamma distribution computer program and graph paper. NOAA Technical Report EDS 11, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, DC, 92 pp. Crutcher, Harold L., 1975: A note on the possible misuse of the Kolmogorov- Smirnov test. Journal of Applied Meteorology , 14 (8), 1600-1603. Crutcher, Harold L., and Ful bright, Danny (National Climatic Center, Environ- mental Data Service, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Asheville, NC) 1974: Specifications of a program for the determination of quantiles of the gamma distribution. 13 pp. (unpublished manuscript). Elderton, William Pal in, 1953: Frequency Curves and Correlation . Harren Press, Washington, DC, 4th ed., 272 pp. Falls, Lee W., 1971: A computer program for standard statistical distribu- tions. NASA Technical Memorandum X-64588, George C. Marshall Space Flight Center, Huntsville, AL, 86 pp. Fisher, N. I., 1973: A note on the evaluation of the incomplete gamma function. Journal of Statistical Computation and Simulation , 2 (4), 325-332. Fisher, Ronald A., 1922: On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London , Series A, Vol. 222, England, 309-368. Friedman, Don G., and Janes, Byron, E., 1957: Estimation of rainfall proba- bilities. Storrs Agricultural Experiment Station Bulletin 332, College of Agriculture, University of Connecticut, Storrs, CT, 22 pp. 50 Greenwood, J. Arthur, and Durand, David, 1960: Aids for fitting the gamma distribution by maximum likelihood. Technome tries , 2 (1), 55-65. Gringorten, Irving I., 1963: A plotting rule for extreme probability paper. Journal of Geophysical Research , 68 (3), 813-814. Gupta, Shanti S., 1960: Order statistics from the gamma distribution. Technometrics , 2 (2), 243-262. Gupta, Shanti S., and Groll, Phyllis A., 1961: Gamma distribution in accept- ance sampling based on life tests. Journal of the American Statistical Association , 56 (296), 942-970. Haggard, William H., Bilton, Thaddeus H., and Crutcher, Harold L., 1973: Maximum rainfall from tropical cyclone systems which cross the Appalachians. Journal of Applied Meteorology , 12 (1), 50-61. Hahn, Gerald J., and Shapiro, Samuel S., 1968: Statistical Models in Engineering . John Wiley & Sons, Inc., New York, NY, 355 pp. Harter, H. Leon, 1964: New Tables of the Incomplete Gamma-Function Ratio and of Percentage Points of the Chi-Square and Beta Distributions . Aerospace Research Laboratories, Office of Aerospace Research, U.S. Air Force, Wright-Patterson Air Force Base, OH, 245 pp. Harter, H. Leon, 1969: A new table of percentage points of the Pearson Type III distribution. Technometrics , 11 (1), 177-187. Hartley, H. 0., and Lewish, W. T., 1959: Fitting of the data to the two parameter gamma distribution with special reference to rainfall data. 650 Program No. 6.008ISU, Statistical Laboratory, Iowa State University, Ames, IA. Hastings, Cecil, Jr. (assisted by Jeanne T. Hayward and James P. Wong, Jr.), 1955: Approximations for Digital Computers . Princeton University Press, Princeton, NJ, 201 pp. Jalickee, J., Sullivan, J., and Rozett, R., 1975: Validation, compaction, and analysis of large environmental data sets. Environmental Data Service , U.S. Department of Commerce, Environmental Data Service, Washington, DC, May issue, 3-9. Kao, John H. K. , 1968: Gamma and Weibull life quality plots. Proceedings of the 19th Annual Institute Conference and Convention , 245-251 . Kenny, John F. , and Keeping, E. S., 1951: Mathematics of Statistics , Part Two . D. van Nostrand Co., Inc., New York, NY, 2nd ed., 429 pp. 51 Kimball, Bradford F., 1960: On the choice of plotting positions on proba- bility paper. Journal of the American Statistical Association , 55 (291), 546-560. Kolmogorov, A. N., 1933: Sulla Determinazione Empirica di una Legge di Distribuzione (On the empirical determination of a distribution law). Giornale dell'Istituto Italiano Degli Attuari , 4, Rome, Italy, 83-91. Lancaster, H. 0., 1969: The Chi-Squared Distribution . John Wiley & Sons, Inc., New York, NY, 356 pp. Lilliefors, Hubert W., 1967: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association , 62 (318), 399-402. Lilliefors, Hubert W., 1969: On the Kolmogorov-Smirnov test for the expo- nential distribution with mean unknown. Journal of the American Statistical Association , 64 (325), 387-389. Lilliefors, Hubert W. (Department of Statistics, George Washington University, Washington, DC), May 1, 1972 (personal communication). Lilliefors, Hubert W., 1973: The Kolmogorov-Smirnov and other distance tests for the gamma distribution and for the extreme-value distribution when parameters must be estimated. George Washington University, supported in part by U.S. Department of Commerce Contract 1-35214, 12 pp. plus tables. Masuyama, M. , and Kuroiwa, Y., 1951: Table for the likelihood solutions of gamma distribution and its medical applications. Statistical Application Research , 1 (1), 18-23. Michal, Aristotle, D., 1947: Matrix and Tensor Calculus . John Wiley & Sons, Inc., New York, NY, 111 pp. Milton, R. C, and Hotchkiss, R., 1969: Computer evaluation of the normal and inverse normal distribution functions. Technometrics , (11 (4), 817-822. Mooley, Diwakar Atmaram, and Crutcher, Harold Lee, 1968: An application of the gamma distribution function to Indian rainfall. ESSA Technical Report EDS 5, Environmental Science Services Administration, U.S. Department of Commerce, Silver Spring, MD, 47 pp. Mooley, Diwakar Atmaram, 1973: Gamma distribution probability model for Asian summer monsoon monthly rainfall. Monthly Weather Review , 101 (2), 160-176. 52 Mooley, Diwakar Atmaram (Indian Institute of Tropical Meteorology, Poona-5, India), 1974 (personal communication). Pearson, E. S., and Hartley, H. 0. (Editors), 1954: Biometrika Tables for Statisticians , Volume I. Cambridge University Press for the Biometrika Trustees, England, 238 pp. Pearson, Karl, 1916: Mathematical contributions to the theory of evolution. -- XIX. Second supplement to a memoir on skew variation. Philosophical Transactions of the Royal Society of London , Series A, Vol. 216, England, 429-457. Pearson, Karl (Editor), 1922: Tables of the Incomplete r-Function . Her Majesty's Stationery Office, London, England, 164 pp. Pearson, Karl (Editor), 1957: Tables of the Incomplete r-Function . Cambridge University Press for the Biometrika Trustees, England, 164 pp. Perl is, Sam, 1952: Theory of Matrices . Addi son-Wesley Publishing Co., Inc., Reading, PA, 83 pp. Pinkham, R. S., 1962: An approximation to the probability integral of the gamma distribution for small values of the shape parameter. Biometrika , 49, 276-278. Pitman, E. J. G. , 1938: The estimation of the location and scale parameters of a continuous population of any given form. Biometrika , 30, 391-421. Roy, S. N. , Gnanadesikan, R., and Srivastava, J. N., 1971: Analysis and Design of Certain Quantitative Multi response Experiments . Pergamon Press, New York, NY, 304 pp. Salvosa, Luis R., 1930: Tables of Pearson's Type III function. Annals of Mathematical Statistics , 1 (2), 191-198 plus appendix. Schickedanz, Paul T., and Krause, Gary F., 1970: A test for the scale parameters of two gamma distributions using the generalized likelihood ratio. Journal of Applied Meteorology , 9 (1), 13-16. Shenton, L. R. , and Bowman, K. 0., 1970: Remarks on Thorn's estimators for the gamma distribution. Monthly Weather Review , 98 (2), 154-160. Smirnov, N., 1936: Sur la Distribution de w 2 (On the distribution of omega squared) . Comptes Rendus Hebdomadaires des Seances d 1 'Academie des Sciences , 202, Paris, France, 449-452. Smirnov, N., 1948: Table for estimating the goodness of fit of empirical distribution. Annals of Mathematical Statistics , 19, 279-281. 53 Southworth, R. W. s and Deleeuw, Samuel L., 1965: Digital Computation and Numerical Methods . McGraw-Hill Book Co., New York, NY, 508 pp. Tate, M. W. , and Hyer, L. A., 1973: Inaccuracy of X 2 test of goodness of fit when expected frequencies are small . Journal of the American Statistical Association , 68 (344), 836-841. Thorn, Herbert C. S., 1947: A note on the gamma distribution. Statistical Laboratory, Iowa State College, Ames, IA. 14 pp. (unpublished manu- script). Thorn, Herbert C. S., 1951: A frequency distribution for precipitation (abstract). Bulletin of the American Meteorological Society , 32 (10), p. 397. Thorn, Herbert C. S., 1957: A statistical method of evaluating augmentation of precipitation by cloud seeding. Technical Report No. 1, U.S. Advisory Committee on Weather Control, Washington, P r . 62 pp. Thorn, Herbert C. S., 1958: A note on the gamma distribution. Monthly Weather Review , 86 (4), 117-122. Thorn, Herbert C. S., 1968: Direct and inverse tables of the gamma distribu- tion. ESSA Technical Report EDS-2, Environmental Science Services Administration, U.S. Department of Commerce, Silver Spring, MD, 30 pp. Thorn, Herbert C. S., and Vestal, Ida B., 1968: Quantiles of monthly precipi- tation for selected stations in the contiguous United States. ESSA Technical Report EDS 6, Environmental Science Services Administration, U.S. Department of Commerce, Silver Spring, MD, 5 pp. plus tables. Wilk, M. B., Gnanadesikan, R., and Huyett, Marilyn J., 1962: Probability plots for the gamma distribution. Technometrics , 4 (1), 1-20. Williams, J. D., 1946: An approximation to the probability integral. Annals of Mathematical Statistics , 17, 363-365. Woodward, W. A., and Gray, H. L., 1975: Minimum variance unbiased estimates in the gamma distribution. Communications in Statistics , 4 (10), 907-922 54 AUTHOR INDEX Anderson, C. W., 9 Andrews, Fred C. , 10 Barger, Gerald L., 5, 6, 7, 9, 10, 35, 44 Bark, L. Dean, 29 Bhattacharjee, G. P., 26 Bilton, Thaddeus H. , 3 Birnbaum, Z. W., 4 Blischke, Wallace R., 7, 8 Blom, Gunnar, 45 Bowman, K. 0. , 8 Campbell, G. A., 3 Chapman, Douglas G., 5, 7 Cohen, A. Clifford, 4 Crutcher, Harold L., 1, 3, 5, 6, 8, 9, 10, 29, 35, 40, 44, 45 Deleeuw, Samuel L., 15 Durand, David, 5, 7 Elderton, William P., 7, 29 Falls, Lee W., 10 Fisher, N. I., 26 Fisher, Ronald A., 5, 8 Friedman, Don G. , 5 Gnanadesikan, R., 15, 21, 44 Gray, H. L., 9 Greenwood, J. Arthur, 5, 7 Gringorten, Irving I., 32, 45 Groll, Phyllis A., 45 Gupta, Shanti S. , 5, 45 Haggard, William H., 3 Hahn, Gerald J., 6, 10 Harter, H. Leon, 4, 44 Hartley, H. 0., 3, 5 Hastings, Cecil, Jr., 7, 18, 19, 20 Hofman, Larry B. , 29 Hotchkiss, R., 18 Huyett, Marilyn J., 15, 21, 44 Hyer, L. A., 39 Jalickee, J., 24 Janes, Byron E. , 5 Kao, John H. K. , 3, 45, 46, 1-24 Keeping, E. S., 10 Kenny, John F. , 10 Kimball, Bradford F., 45 Kolmogorov, A. N., 40 Krause, Gary F. , 5 Kuroiwa, Y., 10 Lancaster, H. 0. , 10 Lewish, W. T. , 5 Lilliefors, Hubert W., 40 McKay, Grady F., 35, 44 Masuyama, M. , 10 Michal , Aristotle, 15 Milton, R. C, 18 Mooley, D. A., 5, 9, 10, 22, 26, 27 Pearson, E. S., 3 Pearson, Karl, 3, 6, 13, 20, 26, 29, 44 Perl is, Sam, 15 Pinkham, R. S., 12, 13, 14, 15 Pitman, E. J. G., 7 Ray, W. D., 9 Roy, S. N., 6 Salvosa, Luis R., 4 Saunders, S. C. , 4 Schickedanz, Paul T., 5 Shapiro, Samuel S., 6, 10 Shenton, L. R. , 8 Smirnov, N., 32, 40 Southworth, R. W., 15 Tate, M. W., 39 Thorn, Herbert C. S., 5, 6, 8, 9, 10, 12, 13, 14, 26, 28, 29, 44, 48 Vestal, Ida B., 5 Wilk, M. B., 5, 6, 9, 10, 13, 15, 21, 22, 44 Williams, J. D., 12, 13 Woodward, W. A., 9 55 APPENDIX 1 TABLES Table 1 Values of h as a function of gamma (7) and selected probabilities .01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .001 1.6658 1.6084 1.5514 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 1 .2807 .003 1.6658 1 . 6084 1.5515 .5010 1.4554 1.4142 1.3766 1.3422 1.3103 1 .2808 .005 1.6658 1.6084 1.5515 .5010 1.4554 1.4142 1.3766 1.3422 1.3103 1 .2808 .006 1.6658 1.6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2808 .007 1.6658 1.6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2808 .008 1.6658 1 . 6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2808 .009 1.6658 1.6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2808 .010 1.6658 1 . 6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2808 .015 1.6658 1 . 6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2809 .020 1.6658 1 . 6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3103 .2809 .025 1.6658 1 . 6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3104 .2810 .030 1.6658 1.6084 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3104 .2810 .035 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3104 .2811 .040 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3104 .2811 .045 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3104 .2812 .050 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3105 .2812 .055 1.6658 1 . 6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3105 .2813 .060 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3105 .2813 .065 1.6658 1 . 6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3106 .2814 .070 1.6658 1 . 6083 1.5515 .5009 1.4554 1.4142 1.3766 1.3422 1.3106 .2814 .075 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3767 1.3423 1.3107 .2815 .080 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3767 1.3423 1.3107 .2815 .085 1.6658 1 . 6083 1.5515 .5009 1.4554 1.4142 1.3767 1.3423 1.3107 .2816 .090 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3767 1.3423 1.3108 .2816 .095 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3767 1.3423 1.3108 .2817 .100 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3767 1.3424 1.3108 .2817 .150 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3768 1.3426 1.3112 .2824 .200 1.6658 1.6083 1.5515 .5009 1.4554 1.4142 1.3769 1.3428 1.3117 .2830 .250 1.6658 1.6083 1.5515 .5008 1.4553 1.4142 1.3770 1.3431 1.3122 .2838 .300 1.6658 1.6083 1.5515 .5008 1.4552 • 1.4142 1.3771 1.3435 1.3127 .2845 .350 1.6658 1.6083 1.5514 .5006 1.4551 1.4142 1.3773 1.3439 1.3134 .2854 .400 1.6658 1.6083 1.5513 .5005 1.4550 1.4142 1.3775 1.3443 1.3140 .2863 .450 1.6658 1 . 6082 1.5512 .5002 1.4548 1.4142 1.3778 1.3448 1.3148 .2873 .500 1.6658 1.6082 1.5510 .4999 1.4546 1.4142 1.3780 1.3453 1.3156 .2883 .550 1.6658 1.6082 1.5507 .4995 1.4543 1.4142 1.3783 1.3459 1.3165 .2896 .600 1.6658 1.6081 1.5502 .4990 1.4540 1.4142 1.3787 1.3467 1.3176 .2909 .631 1.6658 1 . 6080 1.5499 .4990 1.4538 1.4142 1.3789 1.3471 1.3183 .2919 .650 1.6658 1.6079 1.5496 .4984 1.4536 1.4142 1.3791 1.3475 1.3187 .2925 .700 1.6658 1.6076 1.5487 .4975 1.4532 1.4142 1.3796 1.3484 1.3201 .2942 .750 1.6658 1.6069 1.5473 .4964 1.4526 1.4142 1.3801 1.3495 1.3217 .2962 .800 1.6658 1.6057 1.5454 .4949 1.4518 1.4142 1 .3808 1.3508 1.3236 .2987 .850 1.6658 1.6033 1.5426 .4929 1.4509 1 .4142 1.3817 1.3525 1.3260 .3018 .900 1.6658 1.5987 1.5382 .4900 1.4494 1.4142 1.3830 1.3549 1.3293 .3059 .910 1.6658 1.5973 1.5370 .4892 1.4491 1.4142 1.3833 1.3555 1.3302 .3070 .930 1.6658 1.5937 1.5341 .4874 1.4482 1.4142 1.3840 1.3569 1.3321 .3094 .950 1.6656 1 . 5884 1.5301 .4849 1.4471 1 .4142 1 .3850 1.3587 1.3347 .3126 .970 1.6643 1.5799 1.5242 .4813 1.4455 1.4142 1.3864 1.3613 1.3383 .3171 .980 1.6615 1.5732 1.5196 .4786 1.4442 1.4142 1.3874 1.3632 1.3410 .3205 .990 1.6522 1 .5619 1.5123 .4743 1.4423 1.4142 1.3891 1.3662 1.3452 .3257 .995 1 . 6388 1.5514 1.5057 .4703 1.4405 1.4142 1.3905 1.3689 1.3490 .3305 .997 1.6277 1.5443 1.5012 .4678 1.4394 1.4142 1.3915 1.3707 1.3516 .3338 .999 1.6040 1.5309 1.4928 .4629 1.4372 1.4142 1.3933 1.3742 1.3565 .3398 1-1 Table 1 (cont'd) Values of h as a function of gamma (7) and selected probabilities 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 .001 1.2533 1.2277 .2038 .1812 .1600 .1401 .1211 .1033 .0863 1.0702 .003 1.2534 1.2278 .2038 1.1814 .1602 .1403 .1215 .1037 .0868 1.0707 .005 1.2534 1.2278 .2039 .1815 .1604 .1405 .1218 .1040 .0871 1.0711 .006 1.2534 1.2279 .2040 .1815 .1605 .1406 .1219 .1041 .0873 1.0713 .007 1.2534 1.2279 .2040 .1816 .1605 .1407 .1220 .1042 .0874 1.0715 .008 1.2534 1.2279 .2040 .1816 .1606 .1408 .1221 .1044 .0876 1.0716 .009 1.2534 1.2279 .2041 .1817 .1606 .1409 .1222 .1045 .0877 1.0718 .010 1.2535 1.2279 .2041 .1817 .1607 .1409 .2222 .1046 .0878 1.0719 .015 1.2535 1.2281 .2043 .1820 .1611 .1413 .1227 .1051 .0884 1.0725 .020 1.2536 1.2282 .2044 .1822 .1613 .1416 .1230 .1055 .0889 1.0731 .025 1.2537 1.2283 .2046 .1824 .1615 .1419 .1234 .1059 .0893 1.0736 .030 1.2538 1 .2284 .2047 .1826 .1617 .1422 .1237 .1062 .0897 1.0740 .035 1.2538 1.2285 .2049 .1827 .1620 .1424 .1240 U066 .0901 1.0745 .040 1.2539 1.2286 .2050 .1829 .1622 .1427 .1243 .1069 .0905 1.0749 .045 1 . 25^10 1.2287 .2051 .1831 .1624 .1429 .1246 .1072 .0908 1.0752 .050 1.2541 1.2288 .2053 .1833 .1626 .1432 .1248 .1075 .0912 1.0756 .055 1.2541 1 . 2289 .2054 .1834 .1628 .1434 .1251 .1078 .0915 1.0760 .060 1.2542 1.2290 .2056 .1836 .1630 .1436 .1254 .1081 .0918 1.0763 .065 1.2543 1.2291 .2057 .1838 .1632 .1439 .1256 .1084 .0921 1.0767 .070 1.2544 1.2293 .2058 .1839 .1634 .1441 .1259 .1087 .0924 1.0770 .075 1.2545 1.2294 .2060 .1841 .1636 .1443 .1261 .1090 .0927 1.0773 .080 1.2545 1.2295 .2061 .1842 .1638 .1445 .1264 .1092 .0930 1.0776 .085 1.2546 1.2296 .2062 .1844 .1640 .1447 .1266 .1095 .0933 1.0779 .090 1.2547 1.2297 .2064 .1846 .1641 .1449 .1268 .1098 .0936 1.0782 .095 1.2548 1.2298 .2065 .1847 .1643 .1451 .1271 .1100 .0939 1.0785 .100 1.2549 1.2298 .2066 .1849 .1645 .1454 .1273 .1103 .0941 1.0788 .150 1.2557 1.2310 .2079 .1864 .1664 .1474 .1295 .1127 .0968 1.0817 .200 1.2566 1.2321 .2093 .1880 .1680 .1493 .1317 .1150 .0992 1.0843 .250 1.2575 1.2332 .2106 .1895 .1698 .1513 .1338 .1173 .1017 1.0869 .300 1.2585 1.2344 .2121 .1912 .1716 .1532 .1359 .1196 .1041 1.0894 .350 1.2596 1.2357 .2135 .1928 .1735 .1553 .1381 .1219 .1066 1.0920 .400 1.2607 1.2371 .2151 .1946 .1754 .1574 .1404 .1243 .1091 1.0947 .450 1.2620 1.2386 .2168 .1965 .1775 .1596 .1428 .1268 .1118 1.0975 .500 1.2633 1.2402 .2186 .1985 .1797 .1620 .1453 1 .1295 .1146 1 . 1 004 .550 1.2648 1.2419 .2206 .2007 .1820 .1645 .1480 .1324 .1176 1.1035 .600 1.2664 1.2438 .2227 .2030 .1846 .1673 .1509 .1355 .1208 1.1069 .631 1.2664 1.2451 .2241 .2046 ,1863 .1691 .1529 .1375 .1230 1.1091 .650 1.2683 1.2459 .2251 .2057 .1874 .1703 .1541 .1389 .1244 1.1106 .700 1.2704 1.2483 .2278 .2086 .1906 .1737 .1578 .1427 .1283 1.1147 .750 1.2728 1.2511 .2309 .2120 .1943 .1776 .1619 .1470 .1329 1.1194 .800 1.2757 1.2544 .2346 .2161 .1987 .1823 .1668 .1521 .1382 1.1250 .850 1.2793 1.2586 .2392 .2211 .2041 .1880 .1729 .1585 .1448 1.1318 .900 1.2843 1.2642 .2455 .2279 .2114 .1958 .1810 .1670 .1536 1.1409 .910 1.2855 1.2656 .2470 .2296 .2132 .1977 .1830 1 .1691 .1559 1.1432 .930 1.2884 1.2689 .2507 .2335 .2174 .2022 .1877 .1740 .1610 1.1485 .950 1.2921 1.2731 .2553 .2386 .2228 .2079 .1938 .1804 .1676 1.1553 .970 1.2975 1.2792 .2620 .2459 .2306 .2162 .2025 .1894 .1770 1.1651 .980 1.3014 1.2836 .2670 .2512 .2363 .2223 .2089 1 .1961 .1840 1.1723 .990 1.3077 1.2907 .2748 .2597 .2455 .2320 1 .2191 1 .2067 .1950 1.1837 .995 1.3133 1.2971 .2818 .2674 .2537 1 .2407 .2283 1 .2164 .2051 1.1941 .997 1.3171 1.3014 .2866 .2726 .2593 .2467 1 .2346 1 .2229 .2119 1.2012 .999 1.3243 1.3097 1 .2957 .2826 .2700 1 .2581 .2466 1 .2355 .2251 1.2149 1-2 Tabl e 1 (cont'd) Val ues of h as a function of gamma (y) and selected probabilities Y P 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 .001 1.05479 1.04012 1.02614 1.01280 .99996 .98771 .97590 .96460 .95371 .94320 .003 1.05546 1.04093 1.02700 1.01373 .00101 .98881 .97715 .96588 .95515 .94473 .005 1.05592 1.04145 1.02760 1.01437 .00171 .98962 .97796 .96685 .95607 .94575 .006 1.05612 1.04163 1.02783 1.01463 .00200 .98992 .97830 .96713 .95645 .94608 .007 1.05631 1.04190 1.02806 1.01493 .00227 .99021 .97861 .96750 .95679 .94650 .008 1.05648 1.04210 1.02827 1.01512 .00252 .99047 .97889 .96780 .95711 .94680 .009 1.05664 1.04225 1.02847 1.01538 .00275 .99077 .97916 .96808 .95741 .94713 .010 1.05680 1.04245 1.02865 1.01558 .00297 .99096 .97941 .96835 .95768 .94745 .015 .020 1.05749 1.04320 1 .02949 1.01642 .00393 .99197 .98049 .96950 .95888 .94870 1.05809 1.04385 1.03019 1.01717 .00474 .99282 .98140 .97045 .95988 .94975 .025 1.05863 1.04438 1.03082 1.01790 .00546 .99362 .98219 .97123 .96075 .95058 .030 1.05912 1.04495 1.03140 1.01848 .00611 .99427 .93291 .97200 .96153 .95140 .035 1.05959 1.04543 1.03194 1.01903 .00671 .99492 .98358 .97276 .96226 .95221 .040 1.06002 1.04588 1.03244 1 .01960 .00728 .99552 .98420 .97335 .96293 .95285 .045 1.06044 1.04635 1.03292 1 .02008 .00781 .99607 .98479 .97401 .96356 .95356 .050 1 . 06084 1.04681 1.03338 1.02058 .00832 .99656 .98534 .97457 .96416 .95417 .055 1.06123 1.04720 1.03382 1.02103 .00881 .99713 .98588 .95508 .96474 .95473 .060 1.06161 1.047 58 1.03424 1.02150 .00928 .99762 .98639 .97566 .96529 .95531 .065 1.06197 1 .04806 1.03465 1.02192 .00974 .99808 .98689 .97615 .96582 .95585 .070 1.06233 1.04841 1.03505 1.02238 .01018 .99858 .98736 .97663 .96633 .95638 .075 1.06267 1.04873 1.03544 1.02275 .01061 .99897 .98783 .97710 .96683 .95690 .080 1.06301 1.04910 1.03583 1.02313 .01103 .99943 .98828 .97760 .96731 .95740 .085 1.06334 1.04946 1 .03620 1.02353 .01144 .99982 .98873 .97806 .96779 .95791 .090 1.06367 1.04986 1.03656 1.02392 .01184 1.00030 .98916 .97848 .96825 .95833 .095 1.06399 1.05015 1.03692 1.02428 .01223 1 . 00068 .98958 .97895 .96870 .95885 .100 1.06431 1.05051 1.03728 1.02468 .01262 1.00108 .99000 .97935 .96914 .95925 .150 1.06729 1.05366 1.04058 1.02815 .01621 1.00477 .99385 .98331 .97322 .96346 .200 1.07007 1.05655 1.04363 1.03131 .01950 1.00817 .99735 .98691 .97691 .96726 .250 1.07277 1.05941 1.04657 1.03441 .02265 1.01148 1.00069 .99036 .98041 .97081 .300 1.07545 1.06221 1.04948 1.03741 .02576 1.01470 1.00396 .99370 .98383 .97430 .350 1.07817 1.065C1 1.05241 1.04042 .02887 1.01782 1.00724 .99708 .98725 .97783 .400 1.08096 1.06792 1.05540 1.04345 .03205 1.02111 1.01057 1.00046 .99072 .98131 .450 1.08386 1.07092 1.05851 1.04666 .03532 1.02445 1.01400 1.00397 .99429 .98497 .500 1.08690 1 .07408 1.06176 1.05007 .03876 1.02800 1.01759 1.00762 .99801 .98872 .550 1.09014 1.07738 1 .06521 1.05357 .04239 1.03171 1.02138 1.01146 1.00194 .99271 .600 1.09363 1.08098 1.06893 1.05738 .04629 1.03565 1.02545 1.01557 1.00614 .99697 .631 1.09596 1.08343 1.07140 1.05993 .04889 1.03830 1.02814 1.01832 1.00893 .99982 .650 1.09746 1.08498 1 .07299 1.06158 .05055 1.04006 1.02988 1.02012 1.01072 1.00162 .700 1.10172 1.08935 1.07750 1.06618 .05528 1.04486 1.03479 1.02513 1.01580 1.00678 .750 1.10659 1.09433 1.08264 1.07145 .06066 1.05036 1.04038 1.03085 1.02156 1.01264 .800 1.11231 1.10026 1.08869 1.07765 .06698 1.05676 1.04693 1.03745 1.02831 1.01945 .850 1.11938 1.10750 1.09614 1.08526 .07476 1.06472 1.05498 1.04563 1.03661 1.02788 .900 1.12880 1.11723 1.10606 1.09540 .08511 1.07523 1.06571 1.05648 1.04765 1.03903 .910 1.13116 1.11961 1.10854 1.09791 .08770 1.07788 1.06838 1.05925 1.05040 1.04185 .930 1.13662 1.12523 1.11430 1.10381 .09370 1.08398 1.07460 1.06555 1.05681 1.04835 .950 1.14365 1.13245 1.12169 1.11136 .10141 1.09185 1.08259 1.07365 1.06500 1.05665 .970 1 . 1 5368 1.14281 1.13226 1.12216 .11244 1.10305 1.09401 1.08527 1.07679 1.06857 .980 1.16115 1.15043 1.14014 1.13025 .12066 1.11143 1.10253 1.09393 1.08557 1.07748 .990 1.17298 1.16261 1.15262 1.14298 .13370 1.12477 1.11605 1.10766 1.09952 1.09161 .995 1.18374 1.17368 1.16398 1.15465 .14559 1.13686 1.12840 1.12023 1.11226 1.10458 .997 1.19105 1.18122 1.17171 1.16256 .15368 1.14511 1.13681 1.12880 1.12096 1.11340 .999 1.20519 1.19580 1.18669 1.17791 4- 1-. .16940 3 1.16118 1.15318 1.14543 1.13791 1.13058 Table 1 (cont'd) Values of h as a function of gamma (7) and selected probabilities 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 .001 .93315 .92342 .91403 .90497 .89620 .88772 .87951 87156 .86385 .85637 .003 .93478 .92514 .91584 .90685 .89816 .88977 .88164 87377 .86613 .85873 .005 .93580 .92621 .91696 .90802 .89938 .89103 .88294 87511 .86751 .86014 .006 .93622 .92665 .91741 .90850 .89988 .89154 .88347 87565 .86807 .86071 .007 .93660 .92704 .91783 .90892 .90032 .89200 .88394 87613 .86857 .86122 .008 .93694 .92741 .91820 .90932 .90072 .89241 .88437 87657 .86902 .86169 .009 .93727 .92774 .91855 .90968 .90110 .89280 .88477 87698 .86944 .86212 .010 .93757 .92806 .91888 .91002 .90145 .89316 .88514 87737 .86983 .86252 .015 .93888 .92942 .92092 .91147 .90295 .89370 .88672 87899 .87149 .86422 .020 .93966 .93054 .92144 .91266 .90417 .89596 .88801 88031 .87284 .86559 .025 .94090 .93151 .92244 .91369 .90523 .89705 .88912 88144 .87400 .86678 .030 .94174 .93238 .92334 .91462 .90618 .89802 .89011 88246 .87503 .86783 .035 .94251 .93318 .92416 .91546 .90704 .89890 .89102 88338 .87597 .86879 .040 .94323 .93392 .92493 .91624 .90784 .89972 .8918"5 88423 .87684 .86967 .045 .94391 .93461 .92564 .91697 .90859 .90Q49 .89264 88503 .87765 .87049 .050 .94455 .93527 .92632 .91767 .90930 .90121 .89337 88578 .87842 .87127 .055 .94516 .93590 .92696 .91832 .90998 .90190 .89408 88649 .87914 .87201 .060 .94575 .93650 .92758 .91896 .91062 .90256 .89474 88718 .87984 .87271 .065 .94631 .93708 .92817 .91956 .91124 .90319 .89539 88783 .88050 .87339 .070 .94685 .93764 .92874 .92015 .91184 .90379 .89601 88846 .88114 .87404 .07 5 .94738 .93818 .92929 .92071 .91241 .90438 .89661 88907 .88176 .87466 .080 .94789 .93870 .92983 .92126 .91297 .90495 .89719 88966 .88236 .87527 .085 .94839 .93921 .93035 .92180 .91352 .90551 .89775 89023 .88294 .87586 .090 . 94888 .93971 .93086 .92232 .91405 .90605 .89830 89079 .88351 .87644 .095 .94936 .94020 .93136 .92282 .91457 .90658 .89884 89133 .88406 .87700 .100 .94983 .94068 .93185 .92332 .91508 .90709 .89936 89187 .88460 .87754 .150 .95410 .94505 .93631 .92787 .91970 .91179 .90413 89671 .88950 .88251 .200 .95796 .94898 .94032 .93194 .92384 .91599 .90839 90102 .89386 .88692 .250 .96161 .95270 .94409 .93578 .92773 .91993 .91238 90506 .89795 .89104 .300 .96516 .95632 .94777 .93950 .93151 .92376 .91625 90897 .90190 .89503 .350 .96871 .95992 .95142 .94321 .93525 .92755 .92008 21284 .90581 .89897 .400 .97229 .96356 .95511 .94695 .93904 .93138 .92395 91674 .90974 .90294 .450 .97598 .96729 .95890 .95078 .94292 .93530 .92791 92074 .91377 .90700 .500 .97981 .97118 .96284 .95476 .94694 .93937 .93201 92488 .91795 .91121 .550 .98386 .97528 .96699 .95896 .95119 .94365 .93633 92923 .92234 .91563 .600 .98819 .97967 .97142 .96345 .95571 .94822 .94094 93388 .92702 .92034 .631 .99106 .98257 .97436 .96641 .95871 .95124 .94399 93695 .93011 .92346 .650 .99290 .98443 .97624 .96831 .96063 .95318 .94594 93892 .93209 .92545 .700 .99811 .98971 .98158 .97370 .96607 .95866 .95147 94449 .93770 .93110 .750 1.00402 .99569 .98762 .97981 .97223 .96487 .95773 95080 .94405 .93749 .800 1.01095 1 .00269 .99470 .98695 .97944 .97214 .96506 95817 .95147 .94496 .850 1.01946 1.01129 1.00339 .99572 .98828 .98106 .97404 96722 .96058 .95412 .900 1.03077 1.02273 1.01494 1.00738 1.00004 .99291 .98598 97924 .97267 .96629 .910 1.03359 1.02559 1.01782 1.01029 1.00298 .99587 .98897 98224 .97570 .96933 .930 1.04016 1.03222 1.02452 1.01706 1.00980 1.00275 .99589 98922 .98271 .97639 .950 1.04858 1.04074 1.03313 1.02574 1.01856 1.01158 1.00479 99818 .99173 .98546 .970 1.06064 1.05294 1.04545 1.03818 1.03111 1 .02424 1.01754 1 01102 1.00466 .99847 .980 1.06965 1.06204 1.05465 1.04747 1.04049 1.03370 1.02708 1 02063 1.01432 1.00820 .990 1.08397 1.07653 1.06930 1.06227 1.05543 1.04877 1.04227 1 03594 1.02973 1.02373 .995 1.09707 1.08980 1.08272 1.07583 1.06912 1.06259 1.05621 1 04999 1.04387 1.03800 .997 1.10602 1.09886 1.09189 1.08511 1.07849 1.07205 1.06576 1 05962 1.05355 1.04778 .999 1.12347 1.11654 1.10980 1.10322 1.09680 1.09055 1.08443 1 07846 1.07239 1.06691 4- 1-4 Table 1 (cont'd) Values of h as a function of gamma (7) and selected probabilities 4.0 4.1 4.2 4.3 4.4 .001 .84912 .84207 .83522 .82856 .82208 .003 .85154 .84456 .84777 .83118 .82476 .005 .85299 .84605 .83930 .83274 .82635 .006 .85358 .84664 .83991 .83336 .82698 .007 .85410 .84718 .84045 .83391 .82755 .008 .85458 .84767 .84095 .83442 .82807 .009 .85502 .84811 .84141 .83489 .82854 .010 .85542 .84853 .84183 .83532 .82898 .015 .85716 .85030 .84363 .83715 .83084 .020 .85856 .85173 .84509 .83863 .83234 .025 .85977 .85295 .84633 .83989 .83363 .030 .86084 .85404 .84744 .84101 .83476 .035 .86181 .85503 .84844 .84203 .83579 .040 .86271 .85594 .84936 .84297 .83674 .045 .86354 .85679 .85023 .84384 .83763 .050 .86433 .85759 .85104 .84466 .83846 .055 .86508 .85835 .85181 .84544 .83924 .060 .86580 .85907 .85254 .84618 .83999 .065 .86648 .85977 .85324 .84689 .84071 .070 .86714 .86043 .85392 .84758 .84140 .075 .86771 .86108 .85457 .84823 .84207 .080 .86839 .86170 .85520 .84887 .84271 .085 .86899 .96231 .85581 .84949 .84334 .090 .86957 .86290 .85641 .85009 .84395 .095 .87014 .86347 .85699 .85068 .84454 .100 .97069 .86403 .85755 .85125 .84512 .150 .87571 .86911 .86268 .85643 .85035 .200 .88016 .87361 .86722 .86101 .85496 .250 .88433 .87781 .87146 .86528 .85926 .300 .88835 .88186 .87555 .86940 .86340 .350 .89233 .88587 .87958 .87346 .86749 .400 .89633 .88990 .88364 .87754 .87160 .450 .90042 .89402 .88778 .88171 .87579 .500 .90466 .89828 .89207 .88603 .88013 .550 .90911 .90276 .89658 .89055 .08468 .600 .91385 .90753 .90138 .89538 .88953 .631 .91699 .91068 .90455 • .89856 .89273 .650 .91899 .91270 .90658 .90060 .89478 .700 .92467 .91841 .91231 .90637 .90057 .750 .93110 .92487 .91881 .91289 .90712 .800 .93861 .93242 .92639 .92051 .91478 .850 .94782 .94168 .93570 .92986 .92416 .900 .96005 .95398 .94806 .94228 .93663 .910 .96311 .95706 .95115 .94537 .93974 .930 .97022 .96419 .95832 .95258 .94698 .950 .97934 .97337 .96754 .96184 .95628 .970 .99243 .98653 .98076 .97513 .96963 .980 1.00222 .99637 .99066 .98508 .97962 .990 1.01784 1.01209 1.00647 1.00097 .99559 .995 1 .03220 1.02654 1.02101 1.01559 1.01029 .997 1 .04205 1.03646 1.03098 1.02562 1 .02038 .999 1.06133 1.05586 1.05050 1.04526 1.04012 1-5 Table 2 Quantiles for selected probabilities and shape parameter values y .01 ,02 ,03 ,04 ,05 p\ 01 .12394072E- -66 .57987583E- -50 .58446321E- -40 05 .50686677E- -65 .24672558E- -43 .17281647E- -32 .55738755E- -26 10 .57068125E- -50 .26702220E- -33 .57987583E- -25 .58424026E- -20 15 .23014364E- -82 .36387863E- -41 .19781353E- -27 .14621980E- -20 .19434868E- -16 20 .71758381E- -70 .64252996E- -35 .28898849E- -23 .19457400E- -17 .61285408E- -14 25 .35226860E- -60 .45018812E- -30 .49105858E- -20 .51503319E- -15 .53156618E- -12 30 .29174172E- -52 .40969092E- -26 .21407932E- -17 .49132217E- -13 •20378972E- -10 35 .14443668E- -45 .91158216E- -23 .36487897E- -15 .23175873E- -11 .44475215E- -09 40 .90964555E- -40 .72327819E- -20 .31276180E- -13 .65288212E- -10 .64262409E- -08 45 .11861086E- -34 .26121932E- -17 .15859802E- -11 .12406459E- -08 .67765317E- -07 50 .44655350E- -30 .50686667E- -15 .53155408E- -10 .17281646E- -07 .55738784E- -06 55 .61537807E- -26 .59501510E- -13 .12743937E- -08 .18724158E- -06 .37498381E- -05 60 .36982657E- -22 .46127086E- -11 .23169803E- -07 .16486063E- -05 .21369336E- -04 65 .11071557E- -18 .25238371E- -09 .33393304E- -06 .12194781E- -04 .10594199E- -03 70 .18309525E- -15 .10263500E- -07 .39489687E- -05 .77771144E- -04 .46656369E- -03 75 .18155132E- -12 .32318925E- -06 .39379731E- -04 .43656497E- -03 .18567357E- -02 80 .11531127E- -09 .81451000E- -05 .33860213E- -03 .21953358E- -02 .67819976E- -02 85 .49518290E- -07 .16881528E- -03 .25601O05E- -02 .10069388E- -01 .23155919E- -01 90 .15035712E- -04 .29496744E- -02 .17456650E- -01 .43384135E- -01 .76317114E- -01 95 .33514566E- -02 .45909824E- -01 .11613830E 00 .19189907E 00 .26593230E 00 99 .20720132E 00 .55888579E 00 .77456396E 00 .94529479E 00 .10876274E 01 p .01 ,05 ,10 ,15 ,20 ,25 ,30 ,35 ,40 ,45 ,50 ,55 ,60 .65 .70 .75 .80 .85 .90 .95 .99 Y .06 .27341005E-33 .12198728E-21 .12690570E-16 .10922837E-13 .13202243E-11 .54427028E-10 .11363237E-08 .14834807E-07 .13734548E-06 .97804035E-06 .56621782E-05 .27724943E-04 .11822711E-03 .44897382E-03 .15455444E-02 .48959754E-02 .14484238E-01 .40770497E-01 .11287551E 00 .33627848E 00 .12104562E 01 .07 15924982E-28 .15394469E-18 30746325E-14 10078192E-11 61407545E-10 ,14881630E-08 .20128488E-07 .18205136E-06 .12264525E-05 ,65978622E-05 .29722892E-04 .11599791E-03 .40216371E-03 .12628553E-02 .36483960E-02 .98321906E-02 .25072516E-01 .61642753E-01 . 15104388E 00 .40263162E 00 .13190668E 01 .08 59818684E-25 32655833E-16 18916329E-12 .30059212E-10 10957508E-08 ,17827357E-07 .17412158E-06 .11958808E-05 ,63473040E-05 .27669697E-04 .10327696E-03 •34002214E-03 .10095629E-02 .27501669E-02 .69720686E-02 .16663985E-01 .38075347E-01 .84724849E-01 .18968585E 00 .46520323E 00 .14168493E 01 .09 .36133847E-22 .21098305E-14 .46668657E-11 .42227449E-09 , 10322931E-07 .12319157E-06 .93405608E-06 .51787848E-05 , 22834414E-04 .84523175E-04 .27256324E-03 .78630430E-03 .20700208E-02 .50513369E-02 .11577229E-01 .25230481E-01 .52994412E-01 .10924621E 00 .22817082E 00 .52434795E 00 .15061075E 01 .10 .60730484E-20 .59307113E-13 .60730484E-10 .35020257E-08 .62188019E-07 .57917133E-06 .35860860E-05 , 16753047E-04 .63684214E-04 , 20683000E-03 .59339110E-03 .15404287E-02 .36844507E-02 .82373349E-02 .17427776E-01 .35306358E-01 .69389883E-01 .13466330E 00 .26615455E 00 .58043511E 00 .15884778E 01 -r 1-6 Table 2 (cont'd) Quantiles for selected probabilities and shape parameter values > .01 Y -is .20 .25 .30 .35 .29241836E-13 .65254808E-10 .67496979E- -08 .15022227E-06 .13890538E-05 .05 .13359945E-08 .20392131E-06 .42185754E- -05 .32110347E-04 .13798128E-03 .10 •13572860E-06 .65255163E-05 .67500624E- -04 .32372462E-03 .10004220E-02 .15 .20258804E-05 .49554917E-04 .34179690E- -03 .12515739E-02 .31915788E-02 .20 .13789330E-04 .20885173E-03 .10808857E- -02 .32703395E-02 .72826141E-02 .25 .61041859E-04 .63759263E-03 .26421771E- -02 .68998026E-02 .13844491E-01 .30, .20585310E-03 .15877907E-02 •54913025E- -02 .12726658E-01 .23474091E-01 .35 .57545172E-03 .34371293E-02 .10211699E- -01 .21416918E-01 .36822988E-01 .40 .14026009E-02 .67195668E-02 .17522387E- -01 .33739793E-01 .54633354E-01 .45 .30802355E-02 .12163741E-01 .28308739E- -01 .50606536E-01 .77786140E-01 .50 .62348058E-02 .20746339E-01 .43673802E- -01 .73131136E-01 .10736949E 00 .55 .11827103E-01 .337731O9E-01 .65023053E- -01 .10272545E 00 .14478095E 00 .60 .21298804E-01 .53010603E-01 .94205874E- -01 .14125250E 00 •19188839E 00 .65 .36803540E-01 .80913131E-01 .13375913E 00 .19128463E 00 .25129798E 00 .70 .61608021E-01 .12103759E 00 .18734823E 00 .25656591E 00 .32683073E 00 .75 .10086478E 00 .17885916E 00 .26062600E 00 .34289946E 00 .42444158E 00 .80 .16328679E 00 •26354363E 00 .36308525E 00 .46007389E 00 .55418999E 00 .85 .26537126E 00 .39239766E 00 .51268651E 00 .62662755E 00 .73514138E 00 .90 .44479853E 00 .60490232E 00 .75039287E 00 .88481077E 00 .10106953E 01 .95 .82558744E 00 .10305258E 01 .12101161E 01 ■ .13723499E 01 .15219584E 01 .99 .19305934E 01 .22023049E 01 .24338854E 01 .26394092E 01 .28265841E 01 > Y .40 .45 .50 .55 .60 .01 .74153940E-05 .27439797E-04 .78543929E- -04 .18649439E-03 .38483493E-03 .05 .41465372E-03 .98159722E-03 .19660700E- ■02 .34869415E-02 .56448343E-02 .10 .23488772E-02 .45916553E-02 .78953870E- -02 .12366732E-01 .18060438E-01 .15 .64919025E-02 .11358089E-01 .17882890E- -01 .26076151E-01 .35894098E-01 .20 .13392235E-01 •21678219E-01 .32092377E- ■01 .44516986E-01 . 58803353E-01 .25 .23564946E-01 .35943238E-01 .50765522E- -01 .67788174E-01 .86771720E-01 .30 .37541886E-01 .54586943E-01 .74235931E- -01 .96139061E-01 .11998882E 00 .35 .55.911714E-01 .78122844E-01 .10295006E 00 .12996633E 00 .15882163E 00 .40 .79361887E-01 .10718363E 00 .13749795E 00 .16983328E 00 .20382258E 00 .45 .10873061E 00 .14257143E 00 .17865858E 00 .21650947E 00 .25576405E 00 .50 .14507814E 00 .18532838E 00 .22746821E 00 .27103638E 00 .31570202E 00 .55 .18979205E 00 .23684263E 00 .28532593E 00 .33483270E 00 . 38508138E 00 .60 .24475234E 00 .29901585E 00 .35416315E 00 .40986558E 00 .46590925E 00 .65 .31260631E 00 .37454349E 00 .43672857E 00 .49894020E 00 .56104913E 00 .70 .39725716E 00 .46741396E 00 .53709709E 00 .60621791E 00 .67474797E 00 .75 .50480611E 00 .58387147E 00 .66165185E 00 .73821644E 00 .81365358E 00 .80 .64557102E 00 .73447837E 00 .82118721E 00 .90595089E 00 .98899180E 00 .85 .83910173E 00 •93923492E 00 .10361254E 01 .11302414E 01 •12219601E 01 .90 .11298428E 01 .12435490E 01 .13527717E 01 .14582362E 01 .15605034E 01 .95 .16619620E 01 .17944039E 01 .19207294E 01 .20419986E 01 .21590121E 01 .99 .30000967E 01 .31630125E 01 .33174483E 4- 1-7 01 .34649288E 01 . 36065893E 01 Table 2 (cont'd) Quantiles for selected probabilities and shape parameter values y .65 ,01 .71277933E-03 ,05 .85181609E-02 ,10 .24991261E-01 ,15 .47261915E-01 ,20 .74794705E-01 ,25 .10749592E 00 ,30 .14552276E 00 ,35 .18923209E 00 ,40 .23917969E 00 ,45 .29615107E 00 ,50 .36122400E 00 ,55 .43587450E 00 ,60 .52215250E 00 ,65 .62298069E 00 ,70 .74269022E 00 ,75 .88805467E 00 ,80 .10705016E 01 ,85 .13115878E 01 ,90 .16600159E 01 ,95 .22723894E 01 ,99 .37432990E 01 .70 .12126230E-02 , 12163199E-01 .33145498E-01 .60089280E-01 .92338598E-01 .12976492E 00 .17251980E 00 •22096777E 00 .27568007E 00 .33746366E 00 .40742375E 00 .48707106E 00 ■57850019E 00 ■68469319E 00 ■81006375E 00 ■96150709E 00 .11506443E 01 .13993760E 01 .17571285E 01 .23826200E 01 ,38757387E 01 .75 •19272509E-02 •16616388E-01 •42490486E-01 .74280323E-01 •11129296E 00 •15340753E 00 .20079461E 00 .25384184E 00 .31314598E 00 .37954205E 00 .45416698E 00 ■53856877E 00 ■63488778E 00 .74616519E 00 .87689542E 00 .10340914E 01 .12295606E 01 ■14855333E 01 .18521300E 01 .24900976E 01 .40044516E 01 .80 .28980766E-02 .21897533E-01 .52981822E-01 .89739657E-01 .13152850E 00 .17827535E 00 .23019178E 00 .28770163E 00 .35143535E 00 .42226125E 00 .50135123E 00'' .59029232E 00 .69127126E 00 .80738779E 00 .94321522E 00 .11058806E 01 .13073710E 01 .15702343E 01 .19452584E 01 .25951436E 01 .41298791E 01 ■41635370E-02 .28013140E-01 .64568542E-01 •10637579E 00 .15292912E 00 .20424020E 00 .26058067E 00 .32242151E 00 .39043351E 00 .46552234E 00 .54889726E 00 .64218556E 00 .74762055E 00 .86835984E 00 , 10090538E 01 .11769400E 01 .13841796E 01 .16536264E 01 .20367126E 01 .26980239E 01 •42523845E 01 P .01 .05 ,10 ,15 ,20 .25 ,30 ,35 ,40 ,45 .50 ,55 .60 ,65 .70 .75 .80 ,85 .90 .95 .99 .90 .57581294E-02 .34959469E-01 .77196721E-01 .12410289E 00 .17539136E 00 .23119136E 00 .29185095E 00 .35789736E 00 .43004703E 00 .50924612E 00 .59674305E 00 .69420613E 00 .80391524E 00 .92909497E 00 .10744409E 01 .12473273E 01 .14600763E 01 .17358353E 01 .21266601E 01 .27989611E 01 .43722707E 01 .95 .77119084E-02 .42725146E-01 .90811875E-01 .14284155E 00 .19882333E 00 .25903302E 00 , 32390901E 00 .39404222E 00 .47019900E 00 ■55336848E 00 .64483946E 00 ,74632178E 00 .86014162E 00 .98956968E 00 .11394051E 01 .13170975E 01 .15351395E 01 .18169687E 01 .22152434E 01 .28981436E 01 •44897934E 01 1.0 .10050336E-01 ■51293294E-01 •10536052E 00 .16251893E 00 ■22314355E 00 .28768207E 00 .35667494E 00 .43078292E 00 .51082562E 00 •59783700E 00 .69314718E 00 ■79850770E 00 .91629073E 00 .10498221E 00 .12039728E 00 ■13862943E 01 ■16094379E 01 .18971200E 01 .23025851E 01 .29957323E 01 .46051702E 01 1.1 ■15960789E-01 ■70751765E-01 .13705463E 00 .20442986E 00 .27416759E 00 .34712201E 00 ■42406562E 00 .50581289E 00 .59329750E 00 .68764683E 00 •79027528E 00 .90301779E 00 •10283371E 01 .11696666E 01 ■13320160E 01 .15231332E 01 .17559741E 01 .20547908E 01 .24739531E 01 , 31866641E 01 ,48302094E 01 1.2 .23608723E-01 .93145199E-01 .17189837E 00 .24937278E 00 .32797593E 00 .40903367E 00 .49357816E 00 .58259093E 00 .67712632E 00 .77840825E 00 .88793621E 00 .10076279E 00 .11400344E 01 .12886927E 01 .14587463E 01 .16581295E 01 .19000888E 01 .22093856E 01 .26414604E 01 .3372S633E 01 .50486095E 01 -J- 1-8 Table 2 (cont'd) Quantiles for selected probabilities and shape parameter values PV 01 1.3 .33058070E- -01 1.4 .44330292E- -01 1.3 .57415901E-01 05 .11827039E 00 •14592651E 00 .17592316E 00 10 .20955619E 00 .24973648E 00 .29218719E 00 15 .29696530E 00 .34689081E 00 .39888572E 00 20 .38418046E 00 .44246888E 00 .50258701E 00 25 .47304661E 00 .53886879E 00 .60626645E 00 30 .56487458E 00 •63769285E 00 .71182612E 00 35 .66082148E 00 .74027900E 00 .82078780E 00 40 .76206620E 00 .84793267E 00 .93458420E 00 45 .86993056E 00 .96207392E 00 .10547333E 01 50 .98599905E 00 .10843715E 01 .11829869E 01 55 .11122708E 01 .12169045E 01 .13215026E 01 60 .12513831E 01 .13623940E 01 .14730830E 01 65 .14069732E 01 .15245743E 01 .16415562E 01 70 .15843122E 01 •17088396E 01 .18324354E 01 75 .17915140E 01 .19234761E 01 .20541725E 01 80 .20421025E 01 .21822736E 01 .23208138E 01 85 .23613257E 01 .25109475E 01 .26585239E 01 90 •28056478E 01 .29669437E 01 .31256943E 01 95 .35544295E 01 .37325150E 01 .39073640E 01 99 .52613106E 01 .54690535E 01 .56724334E 01 4- 1-9 .10E-25 .10E-24 .10E-23 .10E-22 .10E-21 .10E-20 .10E-19 .10E-18 .10E-17 .10E-16 .10E-15 .10E-14 •10E-13 •10E-12 .10E-11 .10E-10 .10E-09 .10E-08 .10E-07 .10E-06 .10E-05 .10E-04 .10E-03 •10E-02 .10E-01 .10E 00 .12E 00 .1AE 00 .16E 00 .18E 00 .20E 00 .22E 00 .2AE 00 .26E 00 .28E 00 ,30E 00 ,32E 00 ,3AE 00 .36E 00 ,38E 00 .AOE 00 ,A2E 00 ,AAE 00 ,A6E 00 ,A8E 00 ,50E 00 .52E 00 ,5AE 00 ,56E 00 ,58E 00 Table 3 Probabilities for selected quantiles and shape parameter values .01 .02 .03 .OA .05 .06 .07 .5527 .5656 .5787 .5922 .6060 .6201 .63A6 .6A93 .66A5 .6799 .6958 ,7120 .7286 .7A55 .7629 .7807 .7989 ,8175 .8365 .8560 .8759 .8963 .9172 ,9386 ,9603 ,9819 ,9835 .98A8 .9859 .9869 .9878 .9885 .9892 .9899 • 990A .9909 .991A ,9918 .9922 ,9926 .9930 ,9933 .9936 .9939 .9941 .99AA .9946 .99A8 .9950 ,305A ,3198 .3349 ,3506 ,3672 ,3845 ,4026 ,4216 ,4414 ,4622 ,4840 ,5068 ,5307 ,5557 .5819 ,6094 .6381 ,6681 ,6996 ,7326 ,7671 ,8033 ,8411 ,8808 .9221 ,9639 .9671 .9697 ,9719 ,9739 ,9756 ,9771 .9785 .9797 ,9808 .9818 .9828 .9836 .9844 .9852 .9859 .9865 ,9871 ,9877 ,9882 .9887 .9892 .9896 .9900 .9904 ,1687 ,1808 ,1937 ,2076 .2224 .2383 .2554 .2737 ,2932 .3142 .3367 .3607 .3865 .4142 .4438 .4756 .5096 .5460 .5851 .6269 .6717 .7198 ,7713 .8264 .8853 .9A62 .9508 .95A7 .9580 .9609 .9634 .9657 .9677 .9695 .9712 .9727 .9741 ,9754 .9766 .9777 ,9787 .9797 .9806 .9814 .9822 .9830 .9837 .9843 .9849 .9855 .0932 .1022 .1121 .1229 .1347 ,1477 .1620 .1776 .1947 .2135 .2341 .2567 .2815 .3087 .3384 .3711 .4069 .4461 .4892 .5364 .5881 .6449 .7071 .7753 .8498 .9286 .9347 .9398 .9442 .9480 .9513 .9543 .9569 .9593 .9615 .9635 .9654 .9671 .9687 .9701 .9715 .9728 .9740 .9751 .9762 .9772 .9781 .9790 .9798 .9806 .0515 .0578 .0648 .0727 .0816 .0916 .1027 .1153 .1293 .1451 .1628 .1827 .2050 .2300 .2580 .2895 .3248 .3645 .4089 .4588 .5148 .5776 .6481 .7272 .8156 .9113 .9188 .9250 .9304 .9351 .9392 .9429 .9462 .9492 .9519 .9544 .9567 .9588 .9608 .9626 .9643 .9659 .9674 .9688 .9701 .9713 .9725 .9736 .9746 .9756 ,0284 ,0326 ,0375 ,0430 ,0494 ,0567 .0651 .0748 .0859 .0986 .1132 .1300 .1492 .1713 .1967 .2258 .2593 .2977 .3418 .3925 .4506 .5174 .5940 ,6820 .7826 .8941 .9030 .9104 ,9167 ,9223 .9272 .9315 .9354 .9390 .9422 .9452 .9479 .9505 .9528 .9550 .9570 ,9589 .9607 .9624 .9639 .9654 .9668 ,9682 .9694 .9706 ,0157 ,0184 ,0217 .0255 .0299 .0351 .0413 .0485 .0570 .0670 .0787 .0924 .1086 .1276 .1499 .1761 .2069 .2431 .2857 .3356 .3943 .4633 .5443 .6395 .7509 .8772 .8873 .8958 .9031 .9095 .9151 .9202 .9247 .9288 .9326 .9360 .9392 .9421 .9448 .9473 .9497 .9519 ,9540 .9559 ,9578 .9595 .9611 .9627 .9641 .9655 .08 .0087 .0104 .0125 .0151 .0181 .0218 .0262 .0315 .0378 .0455 .0547 .0657 .0790 .0950 .1142 .1374 .1651 .1985 .2387 .2870 .3450 .4148 .4987 .5995 .7203 .8604 .8718 .8814 .8896 .8968 .9032 .9089 .9140 .9186 .9229 .9268 .9304 .9337 .9368 .9397 .9424 .9449 .9472 .9494 .9515 .9535 .9554 .9571 .9588 .9604 .09 .0048 .0059 .0072 .0089 .0110 .0135 .0166 .0204 .0251 .0309 .0380 .0468 .0575 .0708 .0871 .1071 .1318 .1621 .1994 .2454 .3018 .3714 .4569 .5620 .6909 .8439 .8565 .8671 .8762 .8842 .8913 .8976 .9033 .9085 .9132 .9176 .9216 .9253 .9288 .9320 .9350 .9378 .9404 .9429 .9453 .9475 .9496 .9515 .9534 .9552 1-10 Table 3 (cont'd) Probabilities for selected quantiles and shape parameter values \7 T \ •60E 00 .62E 00 .64E 00 .66E 00 .68E 00 .70E 00 .72E 00 •74E 00 .76E 00 .78E 00 .80E 00 .82E 00 .84E 00 .86E 00 .88E 00 .90E 00 .92E 00 .94E 00 .96E 00 .98E 00 .10E 01 .HE 01 .12E 01 .13E 01 .14E 01 •15E 01 .16E 01 .17E 01 .18E 01 .19E 01 .20E 01 .21E 01 .01 .02 .03 .04 .05 06 .07 .08 .09 9908 .9861 .9814 .9766 .9717 .9668 .9619 .9569 9911 .9866 .9821 .9775 .9728 .9681 .9633 .9585 9915 .9872 .9828 .9783 .9738 .9693 .9647 .9600 9918 .9876 .9834 .9791 .9748 .9704 .9660 .9615 9921 .9881 .9840 .9799 .9757 .9715 .9672 .9629 9924 .9885 .9846 .9806 .9766 .9725 .9684 .9642 9927 .9890 .9852 .9813 .9774 .9735 .9695 .9654 9930 .9894 .9857 .9820 .9782 .9744 .9706 .9667 9932 .9897 .9862 .9826 .9790 .9753 .9716 .9678 9935 .9901 .9867 .9832 .9797 .9762 .9726 .9689 9937 .9904 .9872 .9838 .9804 .977D .9735 .9700 9939 .9908 .9876 .9844 .9811 .9778 .9744 .9710 9941 .9911 .9880 .9849 .9817 .9785 .9753 .9720 9943 .9914 .9884 .9854 .9824 .9793 .9761 .9729 9945 .9917 .9888 .9859 .9830 .9799 .9769 .9738 9947 .9920 .9892 .9864 .9835 .9806 .9777 .9747 9949 .9922 .9896 .9868 .9841 .9812 .9784 .9755 9950 .9925 .9899 .9873 .9846 .9819 .9791 .9763 .9928 .9902 .9877 .9851 .9825 .9798 .9770 .9930 .9906 .9881 .9856 .9830 .9804 .9778 .9932 .9909 .9885 .9860 .9836 .9810 .9785 .9942 .9922 ."9902 .9881 .9860 .9838 .9816 .9951 .9934 .9916 .9898 .9880 .9862 .9843 .9943 .9928 .9913 .9897 .9881 .9865 .9951 .9938 .9925 .9911 .9897 .9883 .9958 .9947 .9935 .9923 .9911 .9899 .9954 .9944 .9934 .9923 .9913 .9951 .9942 .9,950 .9956 .9933 .9942 .9949 .9956 .9924 .9934 .9942 .9950 .9956 ,10 15 20 ,25 30 35 ,40 .45 .50 0026 .0001 0033 .0002 0042 .0003 0053 .0004 0066 .0005 0083 .0008 0105 .0011 .0001 0132 .0015 .0002 0167 .0021 .0003 0210 .0030 .0004 0264 .0043 .0007 .0001 0332 .0060 .0011 .0002 0418 .0085 .0017 .0003 1-11 Table 3 (cont'd) Probabilities for selected quantiles and shape parameter values ,10 .15 20 25 30 .35 .40 .45 .50 .12E 00 . 14E 00 .16E 00 .18E 00 .20E 00 .22E 00 .24E 00 .26E 00 .28E 00 .30E 00 .32E 00 . 34E 00 .36E 00 . 38E 00 .40E 00 .42E 00 .44E 00 .46E 00 .48E 00 . 50E 00 .52E 00 .54E 00 .56E 00 ,58E 00 . 60E 00 . 62E 00 . 64E 00 ,66E 00 ,68E 00 , 70E 00 ,72E 00 ,74E 00 ,76E 00 ,78E 00 ,80E 00 ,82E 00 .0527 .0663 .0835 .1051 .1323 .1666 .2097 .2640 .3324 .4185 .5268 .6626 .8276 .8413 .8529 .8629 .8717 .8794 .8864 .8926 ,8983 ,9036 .9084 ,9128 .9169 ,9207 ,9243 ,9276 ,9307 ,9336 ,9364 ,9390 ,9414 .9437 ,9459 ,9480 ,9500 ,9518 ,9536 ,9553 ,9569 ,9585 ,9599 ,9614 9627 ,9640 ,9652 ,9664 ,9675 .0120 .0170 .0240 .0339 .0479 .0676 .0955 .1349 .1906 .2692 .3802 .5365 .7491 .7680 .7840 .7979 .8101 .8210 .8309 .8398 .8479 .8554 ,8623 .8687 .8746 ,8801 ,8853 ,8902 ,8947 ,8990 ,9031 ,9069 ,9105 .9139 ,9172 ,9202 ,9232 ,9260 ,9286 ,9312 ,9336 ,9359 ,9381 ,9402 ,9422 ,9442 ,9460 ,9478 ,9495 .0027 .0043 .0069 .0109 .0173 .0274 .0434 .0687 .1089 .1726 .2735 .4329 .6760 .6989 .7185 .7356 .7508 .7644 .7768 .7880 .7983 .8077 .8165 .8247 .8323 .8394 .8461 ,8523 ,8582 ,8638 .8691 ,8741 ,8788 .8833 ,8875 ,8916 ,8954 ,8991 ,9026 ,9060 ,9092 ,9123 ,9152 ,9180 ,9207 ,9233 ,9258 9282 ,9305 .0006 .0011 .0020 .0035 .0062 .0110 .0196 .0349 .0620 .1103 .1962 .3482 .6083 .6343 .6567 .6764 .6940 .7099 .7243 .7375 .7496 .7609 .7713 .7811 .7902 .7987 .8068 .8143 .8215 .8282 .8346 .8407 .8465 .8520 .8572 ,8622 .8669 ,8715 .8758 .8800 ,8840 .8878 .8914 .8949 ,8983 ,9015 .9046 .9076 ,9105 .0001 .0003 .0006 .0011 .0022 .0044 .0089 .0177 .0352 .0703 .1402 .2792 .5459 .5740 .5986 .6203 .6398 .6575 .6737 .6885 .7023 .7151 .7270 .7381 .7485 .7584 .7676 .7764 .7847 .7925 .8000 .8071 .8138 .8202 .8264 .8322 .8378 .8432 .8483 .8533 .8580 .8625 .8669 .8710 .8751 .8789 .8826 .8862 .8897 .0001 .0002 .0004 .0008 .0018 .0040 .0089 .0200 .0448 .1000 .2233 .4886 .5182 .5442 .5674 .5884 .6076 .6251 .6414 .6564 .6705 .6836 .6960 .7076 .7185 .7289 .7387 .7480 .7568 .7653 ,7733 ,7809 ,7882 ,7952 ,8019 ,8083 ,8145 .8203 ,8260 ,8314 ,8366 ,8416 ,8465 ,8511 ,8556 ,8599 ,8641 ,8681 .0001 .0003 .0007 .0018 .0045 .0113 .0283 .0711 .1781 .4362 .4667 .4936 .5179 .5399 .5601 .5788 .5961 .6122 .6273 .6415 .6548 .6675 .6794 .6907 .7014 .7117 .7214 .7307 .7395 .7480 .7561 .7639 .7713 ,7785 ,7854 .7919 ,7983 ,8044 .8102 ,8159 ,8213 .8266 ,8316 ,8365 .8412 .8458 .0001 .0003 .0008 .0023 .0063 .0179 .0504 .1417 .3885 .4192 .4467 .4715 .4943 .5152 .5347 .5528 .5698 .5857 .6007 .6149 .6284 .6411 .6532 .6648 .6758 .6863 .6964 .7060 .7152 .7241 .7325 .7407 .7485 .7560 .7633 .7703 .7770 .7835 .7897 .7958 .8016 .8072 .8126 .8179 .8230 .0001 .0004 .0011 .0036 .0113 .0357 ,1125 .3453 .3758 .4033 .4284 .4515 ,4729 .4929 .5116 .5292 .5457 ,5614 .5763 .5904 .6039 .6167 .6289 ,6406 .6518 .6625 ,6728 ,6827 ,6922 ,7013 ,7101 ,7185 .7267 .7345 .7421 7494 7565 7633 7699 7762 7824 7883 7941 ,7997 4- 1-12 Table 3 cont'd) Probabilities for selected quantil es and shape parameter values \ y .10 .15 .20 .25 .30 .35 .40 .45 .50 .84E 00 .9686 .9512 .9327 .9133 .8930 .8719 .8502 .8279 .8051 .86E 00 .9696 .9528 .9348 .9159 .8962 .8757 .8544 .8326 .8103 .88E 00 .9706 .9543 .9369 .9185 .8993 .8793 .8585 .8372 .8154 .90E 00 .9716 .9557 .9388 .9210 .9023 .8827 .8625 .8417 .8203 .92E 00 .9725 .9571 .9407 .9234 .9051 .8861 .8664 .8460 .8250 .94E 00 .9734 .9585 .9425 .9257 .9079 .8894 .8701 .8502 .8297 .96E 00 .9743 .9598 .9443 .9279 .9106 .8925 .8737 .8542 .8341 .98E 00 .9751 .9610 .9460 .9300 .9132 .8955 .8772 .8581 .8385 .10E 01 .9759 .9622 .9476 .9321 .9157 .8985 .8805 .8619 .8427 .HE 01 .9794 .9676 .9550 .9414 .9270 .9118 .895-9 .8792 .8620 .12E 01 .9823 .9722 .9611 .9493 .9366 .9232 .9090 .8942 .8787 ,13E 01 .9848 .9760 .9664 .9560 .9448 .9329 .9203 .9070 .8931 .14E 01 .9869 .9792 .9708 .9617 .9518 .9413 .9301 .9182 .9057 .15E 01 .9887 .9820 .9746 .9666 .9579 .9485 .9386 .9279 .9167 .16E 01 .9902 .9843 .9779 .9708 .9631 .9548 .9459 .9364 .9264 .17E 01 .9914 .9863 .9807 .9744 .9676 .9603 .9523 .9438 .9348 .18E 01 .9926 .9881 .9831 .9776 .9716 .9650 .9579 .9503 .9422 .19E 01 .9935 .9896 .9852 .9803 .9750 .9692 .9628 .9560 .9487 .20E 01 .9943 .9909 .9870 .9827 .9780 .9728 .9671 .9611 .9545 .21E 01 .9950 .9920 .9886 .9848 .9806 .9760 .9709 .9655 .9596 .22E 01 .9930 .9900 .9866 .9829 .9787 .9742 .9694 .9641 .23E 01 .9938 .9912 .9882 .9849 .9812 .9772 .9728 .9680 .24E 01 .9946 .9922 .9896 .9866 .9833 .9797 .9-758 .9715 .25E 01 .9952 .9931 .9908 .9882 .9852 .9820 .9785 .9747 .26E 01 .9939 .9919 .9895 .9869 .9840 .9809 .9774 .27E 01 .9947 .9928 .9907 .9884 .9858 .9830 .9799 •28E 01 .9953 .9936 .9918 .9897 .9874 .9848 .9820 .29E 01 .9943 .9927 .9908 .9888 .9865 .9840 .30E 01 .9950 .9935 .9918 .9900 .9879 .9857 .35E 01 .9964 .9954 .9944 .9932 .9918 .40E 01 .9974 .9968 .9961 .9953 \ y .55 .60 .65 .70 .75 .80 .85 .90 .95 .10E-06 .0002 .0001 .10E-05 .0006 .0003 .0001 .10E-04 .0020 .0011 .0006 .0003 .0002 .0001 .10E-03 .0071 .0045 .0028 .0017 .0011 .0007 .0004 .0003 .0002 .10E-02 .0252 .0177 .0125 .0087 .0061 .0043 .0030 .0021 .0014 .10E-01 .0890 .0704 .0555 .0436 .0343 .0268 .0210 .0164 .0128 .10E 00 .3062 .2709 .2392 .2108 .1855 .1628 .1427 .1249 .1091 .12E 00 .3361 .3000 .2673 .2376 .2109 .1868 .1652 .1458 .1285 .14E 00 .3634 .3267 .2932 .2626 .2347 .2095 .1866 .1660 .1474 .16E 00 .3884 .3514 .3173 .2860 .2573 .2311 .2072 .1854 .1657 •18E 00 .4116 .3744 .3400 .3081 .2788 .2517 .2270 .2043 .1836 .20E 00 .4332 .3960 .3614 .3291 .2992 .2716 .2460 .2226 .2010 .22E 00 .4534 .4164 .3816 .3491 .3188 .2906 .2645 .2403 .2180 .24E 00 .4725 .4356 .4008 .3681 4- 1-13 .3375 .3089 .2823 .2575 .2346 Table 3 (cont'd) Probabilities for selected quantiles and shape parameter values y ,55 ,60 ,65 .70 75 ,80 ,85 ,90 .95 26E 00 .4905 .4538 .4190 .3863 .3555 .3266 .2995 .2743 .2508 28E 00 .5075 .4711 .4365 .4037 .3727 .3436 .3162 .2906 .2666 30E 00 .5237 .4876 .4532 .4204 .3894 .3601 .3324 .3064 .2820 32E 00 .5391 .5033 .4691 .4365 .4054 .3760 .3481 .3218 .2971 34E 00 .5537 .5184 .4844 .4519 .4209 .3914 .3634 .3369 .3118 36E 00 .5677 .5328 .4991 .4668 .4359 .4063 .3782 .3515 .3262 38E 00 .5811 .5466 .5132 .4811 .4503 .4208 .3926 .3658 .3403 40E 00 .5939 .5598 .5268 .4950 .4643 .4349 .4067 .3797 .3541 42E 00 .6062 .5726 .5399 .5083 .4778 .4485 .4203 .3933 .3675 44E 00 .6179 .5848 .5526 .5213 .4910 .4617 .4336 .4066 .3807 46E 00 .6292 .5966 .5647 .5337 .5037 .4746 .4465 .4195 .3935 48E 00 .6401 .6080 .5765 .5458 .5160 .4871 .4591 .4321 .4061 50E 00 .6505 .6189 .5879 .5575 .5279 .4992 .4713 .4444 .4184 52E 00 .6606 .6295 .5988 .5688 .5395 .5110 .4833 .4564 .4305 54E 00 .6703 .6397 .6095 .5798 .5508 .5225 .4949 .4682 .4423 56E 00 .6796 .6495 .6197 .5904 .5617 .5336 .5063 .4796 .4538 58E 00 .6887 .6590 .6297 .6008 .5724 .5445 .5173 .4908 .4651 60E 00 .6974 .6682 .6393 .6108 .5827 .5551 .5281 .5018 .4761 62E 00 .7058 .6771 .6486 .6205 .5927 .5654 .5386 .5124 .4869 64E 00 .7139 .6857 .6577 .6299 .6025 .5754 .5489 .5229 .4975 66E 00 .7217 .6940 .6664 .6391 .6120 .5852 .5589 .5331 .5078 68E 00 .7293 .7021 .6749 .6480 .6212 .5947 .5687 .5431 .5179 70E 00 .7366 .7099 .6832 .6566 .6302 .6040 .5782 .5528 .5279 72E 00 .7437 .7175 .6912 .6650 .6389 .6131 .5875 .5623 .5375 74E 00 .7506 .7248 .6990 .6731 .6474 .6219 .5966 .5716 .5470 76E 00 .7573 .7319 .7065 .6811 .6557 .6305 .6055 .5807 .5563 78E 00 .7637 .7388 .7138 .6888 .6638 .6389 .6141 .5896 .5654 80E 00 .7699 .7455 .7210 .6963 .6716 .6470 .6226 .5983 .5743 82E 00 .7760 .7520 .7279 .7036 .6793 .6550 .6308 .6068 .5831 84E 00 .7819 .7584 .7346 .7107 .6867 .6628 .6389 .6151 .5916 86E 00 .7876 .7645 .7411 .7176 .6940 .6704 .6468 .6233 .5999 88E 00 .7931 .7704 .7475 .7244 .7011 .6778 .6545 .6312 .6081 90E 00 .7984 .7762 .7537 .7309 .7080 .6850 .6620 .6390 .6161 92E 00 .8036 .7818 .7597 .7373 .7147 .6920 .6693 .6466 .6240 94E 00 .8087 .7873 .7655 .7435 .7213 .6989 .6765 .6540 .6317 96E 00 .8136 .7926 .7712 .7496 .7277 .7056 .6835 .6613 .6392 98E 00 .8183 .7977 .7767 .7554 .7339 .7122 .6903 .6684 .6465 10E 01 .8230 .8027 .7821 .7612 .7400 .7186 .6970 .6754 .6537 HE 01 .8442 .8258 .8070 .7878 .7682 .7484 .7283 .7080 .6876 12E 01 .8626 .8459 .8288 .8112 .7932 .7748 .7562 .7372 .7181 13E 01 .8786 .8635 .8479 .8318 .8153 .7984 .7811 .7635 .7456 14E 01 .8926 .8790 .8648 .8501 .8349 .8194 .8034 .7870 .7704 15E 01 .9049 .8926 .8797 .8663 .8524 .8381 .8233 .8082 .7927 16E 01 .9157 .9045 .8928 .8806 .8679 .8548 .8412 .8272 .8128 17E 01 .9252 .9151 .9045 .8934 .8818 .8697 .8572 .8443 .8310 18E 01 .9336 .9244 .9148 .9047 .8941 .8831 .8716 .8597 .8474 19E 01 .9410 .9327 .9240 .9148 .9051 .8950 .8845 .8735 .8622 20E 01 .9475 .9400 .9321 .9237 .9149 .9057 .8961 .8860 .8755 21E 01 .9533 .9465 .9393 .9317 .9237 .9153 .9064 .8972 .8876 1-14 Y Table 3 (cont'd) Probabilities for selected quantiles and shape parameter values ,55 .60 ,65 70 ,75 ,80 ,85 ,90 .95 22E 01 .9584 .9523 .9458 .9389 .9316 .9239 .9158 .9073 .8984 23E 01 .9629 .9574 .9515 .9453 .9386 .9316 .9242 .9164 .9083 24E 01 .9669 .9620 .9566 .9509 .9449 .9385 .9317 .9246 .9171 25E 01 .9705 .9660 .9612 .9560 .9505 .9447 .9385 .9320 .9251 26E 01 .9737 .9696 .9652 .9606 .9556 .9503 .9446 .9386 .9323 27E 01 .9765 .9728 .9689 .9646 .9601 .9553 .9501 .9446 .9389 28E 01 .9790 .9757 .9721 .9683 .9641 .9597 .9550 .9501 .9448 29E 01 .9812 .9782 .9750 .9715 .9678 .9638 .9595 .9549 .9501 30E 01 .9832 .9805 .9776 .9744 .9710 .9674 .9635 .9593 .9549 35E 01 .9904 .9888 .9870 .9851 .9830 .9807 .9783 .9756 .9728 40E 01 .9944 .9935 .9924 .9912 .9900 .9886 .9870 .9854 .9836 45E 01 .9968 .9962 .9956 .9948 .9941 .9932 .9922 .9912 .9901 50E 01 .9970 .9965 .9959 .9954 .9947 .9940 55E 01 .9968 .9964 y 1.0 1.1 1.2 1.3 1.4 1.5 10E-03 .0001 10E-02 .0010 .0005 .0002 .0001 10E-01 .0100 .0060 .0036 .0021 .0013 .0007 10E 00 .0952 .0721 .0542 .0406 .0302 .0224 12E 00 .1131 .0872 .0668 .0509 .0386 .0291 14E 00 .1306 .1022 .0795 .0615 .0473 .0363 16E 00 .1479 .1172 .0923 .0724 .0564 .0438 18E 00 .1647 .1320 .1052 .0834 .0658 .0516 20E 00 .1813 .1468 .1181 .0946 .0754 .0598 22E 00 .1975 .1613 .1311 .1059 .0852 .0681 24E 00 .2134 .1758 .1440 .1173 .0951 .0767 26E 00 .2289 .1900 .1568 .1288 .1052 .0855 28E 00 .2442 .2041 .1696 .1402 .1154 .1945 30E 00 .2592 .2180 .1823 .1517 .1257 .1036 32E 00 .2739 .2317 .1950 .1632 .1360 .1128 34E 00 .2882 .2452 .2075 .1747 .1464 .1221 36E 00 .3023 .2586 .2200 .1862 .1568 .1315 38E 00 .3161 .2717 .2323 .1976 .1673 .1410 40E 00 .3297 .2847 .2445 .2090 .1778 .1505 42E 00 .3430 .2974 .2566 .2203 .1882 .1601 44E 00 .3560 .3100 .2686 .2316 .1987 .1697 46E 00 .3687 .3224 .2804 .2427 .2092 .1794 48E 00 .3812 .3346 .2921 .2539 .2196 .1891 50E 00 .3935 .3466 .3037 .2649 .2299 .1987 52E 00 .4055 .3584 .3151 .2758 .2403 .2084 54E 00 .4173 .3700 .3264 .2866 .2506 .2181 56E 00 .4288 .3814 .3376 .2974 .2608 .2278 58E 00 .4401 .3926 .3486 .3080 .2710 .2374 60E 00 .4512 .4037 .3594 .3186 .2811 .2470 62E 00 .4621 .4146 .3702 .3290 .2912 .2566 64E 00 .4727 .4252 .3807 .3394 .3011 .2661 1-15 Table 3 (cont'd) Probabilities for selected quantiles and shape parameter values 1.0 1.1 1.2 1.3 1.4 1.5 T \ 66K 00 .4831 .4358 .3912 .3496 .3111 .2756 68E 00 .4934 .4461 .4015 .3597 .3209 .2851 70F. 00 .5034 .4563 .4116 .3697 .3306 .2945 72E 00 .5132 .4662 .4216 .3796 .3403 .3038 74E 00 .5229 .4761 .4315 .3894 .3499 .3131 76E 00 .5323 .4857 .4412 .3990 .3594 .3223 78E 00 .5416 .4952 .4508 .4085 .3688 .3315 80E 00 .5507 .5045 .4602 .4180 .3781 .3406 82E 00 .5596 .5137 .4695 .4273 .3873 .3496 84 E 00 .5683 .9997 .4786 .4365 .3964 .3586 86E 00 .5768 .5315 .4876 .4455 .4054 .3675 88E 00 .5852 .5402 .4965 .4545 .4144 .3763 90E 00 .5934 .5487 .5053 .4633 .4232 .3851 92E 00 .6015 .5571 .5139 .4721 .4319 .3937 94 E 00 .6094 .5653 .5223 .4807 .4406 .4023 96E 00 .6171 .5734 .5307 .4891 .4491 .4108 98E 00 .6247 .5814 .5389 .4975 .4576 .4192 10E 01 .6321 .5892 .5470 .5058 .4659 .4276 HE 01 .6671 .6262 .5854 .5453 .5061 .4681 12E 01 .6988 .6599 .6209 .5821 .5439 .5064 13E 01 .7275 .6907 .6536 .6163 .5792 .5425 14E 01 .7534 .7188 .6835 .6479 .6121 .5765 15E 01 .7769 .7444 .7111 .6771 .6428 .6084 16E 01 .7981 .7677 .7363 .7041 .6713 .6382 17E 01 .8173 .7890 .7594 .7290 .6978 .6660 18E 01 .8347 .8083 .7806 .7519 .7223 .6920 19E 01 .8504 .8259 .8000 .7729 .7449 .7161 20E 01 .8647 .8419 .8177 .7923 .7659 .7385 21E 01 .8775 .8564 .8339 .8101 .7852 .7593 22E 01 .8892 .8696 .8487 .8265 .8031 .7786 23E 01 .8997 .8816 .8622 .8414 .8195 .7965 24E 01 .9093 .8926 .8745 .8552 .8346 .8130 25E 01 .9179 .9025 .8858 .8678 .8485 .8282 26E 01 .9257 .9115 .8960 .8793 .8613 .8423 27E 01 .9328 .9197 .9054 .8898 .8731 .8553 28E 01 .9392 .9271 .9139 .8995 .8839 .8672 29E 01 .9450 .9339 .9217 .9083 .8938 .8782 30E 01 .9502 .9400 .9287 .9164 .9029 .8884 35E 01 .9698 .9632 .9557 .9474 .9382 .9281 40E 01 .9817 .9774 .9726 .9670 .9609 .9540 45E 01 .9889 .9862 .9830 .9794 .9753 .9707 50E 01 .9933 .9915 .9895 .9872 .9845 .9814 55E 01 .9959 .9948 .9935 .9920 .9903 .9883 60E 01 .9968 .9960 .9951 .9939 .9926 65E 01 .9962 .9954 4- 1-16 Table 4.1 Algorithm input for quantile ratio computations for 7 = 3 and selected probabilities 22 .68440903E-01 . 13539709EC00 -. 22276772ECO0 .70154770E-01 .14121096EC00 -. 2O739324ECO0 .71286686E-01 . 14473000EC00 -. 1969601 IE £00 .72157844E-01 . 14727630EC00 -. 18882653ECO0 .72877158E-01 . 14927665EG00 - . 18207035ECO0 .73495800E-01 . 1 5092637EC00 -. 17624 132EC00 .74042243E-01 . 15233050ECOO -. 17 ] 07946EC00 .74534106E-01 . 15355322EC00 - . 16642859ECO0 .74983064E-01 . 1 5463582EC00 -. 16? 18 1 18ECO0 .75397306E-01 . 1556O697EC0O -. 15826359E tOO .78466676E-01 . 16201847E COO - . 1293475 1 ECOO .B0601493E-01 .16572263EC00 -. 10949622E COO ,823in8A6E-01 . 16827926Et00 -. 9363 1376E-01 .83770337E-01 . 170 19135EC00 - . 80647535E-01 .85063543E-01 . 17 1687 16EC00 - . 69 1 28 13 1E-01 .86237339E-01 . 17288987E COO -. 588 15663E-01 .87320986E-01 . 1738742 1 E COO - . 4942 1235E-01 .88334038E-01 . 17468793E COO - . 4075 1650E-01 .89290312E-01 . 17536452EC00 -. 32673356E-01 .97161781E-01 .17782912EC0O . 29632223E-01 .10376223EfcOO . l76lo695E£00 .75610992E-01 .11009157EC00 .17164814EC00 . 1 1392607E COO .116633S5EC00 . 16447786E COO . 14736432E COO .123825"59Ecno . 1 539 156 IE COO . l767689oE COO • 13228269Et00 . 13830843E GOO . 201 509 17E COO .14323056EC00 . 1 1 36456 1EC00 . 2 18 15568ECO0 .16034850EC00 . 66692577E-01 . 2 1 1 73993E COO .16282487ECO0 . 59 175 105E-0 1 . 20779 170E COO . 16556126EC00 . 50678327E-01 . 2O259932ECO0 .16862527EC00 .40937174E-01 . 19578777E COO .17211527ECOO . 29562686E-01 . 18677698E COO .17618246EC00 . 1595 1663E-01 . 17467069EC00 .181077B9E£00 -.907867 13E-03 . 15791095E COO .18726705EC00 -. 22914627E-01 . 13350043EC00 • 19578077EC00 -. 5433449 1E-01 . 94509002E-O1 .20984487EC00 -. 10878365EC00 . 17934449E-01 .21193453EC00 -. 1 171 1488EC00 . 54016368E-02 .21425678EC00 -. 126442 14EC00 -. 88664720E-02 .21687243EC00 - . 13702993EC00 - . 25329808E-01 .21987018EC00 - . 14926875EC00 - . 44702873E-01 ■•32197761EC00 -. 13366428EC00PB 1 ■.24387948EC00 -.9078 1245E-01PB 2 ■•19828904EC00 -. 102395 16EC00PB 3 -.16592193EC00 -. 10873253EC00PB 4 ..14089894EC00 - . 98523796E-01PB 5 ■.12043555EC0O -. 82914629E-01PB 6 -.10315707EC00 -.69598948E-01PB 7 ..88146949E-01 -. 59247622E-01 PB 8 ■.74920111E-01 -.45638966E-01PB 9 ■•63128816E-01 -. 3 36647 10E-01PB 10 .14208264E-01 . 52O1O895E-01 PB1 1 .59031710E-01 .10588637EC00PB12 .90242719E-01 . 14o87334ECOOPBl3 .11409116ECOO .16383555EC00PB14 .13310291EC00 . 18 16505 1EC00PB15 .14875102EC00 . 18987157E C00PB16 .16193758EC00 . 19637927EC00PB17 .17320441ECOO . 19992577EC00PB18 .18294635EC0O . 19853831EC00PB19 .23350478EC00 . 12298471E C00PB20 .24196094EC00 . 33448469E-02PB2 1 .22802780EC00 -. 1 1724675EC00PB22 . 19684960EC00 -. 22394455E C00PB23 •14950527EC00 -. 298 164 13E C00PB24 .84872764E-01 -. 32660397E C00PB25 .57899927E-04 -. 27950802EC00PB26 -.1 05745 22 E COO -. 98<,92 325 E -01 PB27 ..11667903EC0O -. 67457 137E-01 PB2 8 -. 1 2736 135E COO -. 37576306E-01PB29 ..13738096ECOO -. 45268249E-02PB30 •.14630390EC00 . 35312979E-01PB3 1 ■.15341020ECOO .75727046E-01PB32 ..1 5695878 E COO . 1 1480029E C00PB33 •.15382489EG00 . 1529648 1EC00PB34 -.13618683EC00 . 1732298 1EC00PB35 -.75570225E-01 . 14698614EC00PB36 ■.63591015E-01 . 14537149EC00PB37 ..49088761E-01 . 12503749EC00PB38 -.31925988E-01 . 10896349EC00PB39 ..10820995E-01 . 885861 13E-01PB40 1-17 Table 4.1 (cont'd) Algorithm input for quantile ratio computations for 7 =3-22 and selected probabilities .2233B678E600 -. 16376294E600 -.68085797E-01 . 1 5607192E-01 .22764993E600 -. 18132426E600 -.97348141E-01 . 50032843E-01 .23308347E600 -.20444933E600 -. 1360261 1E600 .97684746E-01 .24063008E600 -. 2367903 1 E600 - . 19? 129 19E 600 . 16944486E600 .25323921E600 -.2921 1361E600 -.29178592E600 .30643882E600 .56194542E-01PB41 .18868993E-01PB42 -.44241221E-01PB43 •.12315575E600PB44 -.36690261E600PB45 .20352448E602 .20432134E602 .20290268E602 .20151358E602 .20022172E602 .l990«029E602 .l9796472Efi02 .I96985IOE6O2 .196090*1^602 .19527109E602 .19A51746E602 .19382188E602 .19317737E602 .1925787*6602 .192020*2E602 .I9I49829E6O2 .19100868E602 .19054831E602 .l9oilA56E £ o2 .18970A86E602 -.30696131EC01 ■.2A206541EC01 •.18825336E601 ■.14298066EC01 ..10427771E601 ..70710676E600 ..41233718E600 -.13071353E600 •83611659E-01 .29515448E600 •48743608Ee00 .66326397E600 .8249ol66E600 .97419274E600 .11126633E601 .12413842EC01 •13620240EC01 .14748976E601 •13809643E601 .16809039E601 -.26693220E600 -.76621535E-01 .24652304E-01 .78158101E-01 .10398814E600 .11292547E600 •11117399E600 .10250214E600 .89292065E-01 .73O97500E-01 .54965963E-01 .35608267E-01 •15518014E-01 -.49549665E-02 -.25566356E-01 -.A6139671E-01 -.66549905E-01 -.86703487E-01 -.10«»55572E600 -.12605538E600 .60260049E-02 ..3431313AE-02 ..49226591E-02 ..38931192E-02 -.22035723E-02 ■.54602588E-03 .83419816E-03 .18627997E-02 .25372006E-02 .2883250AE-02 .29433733E-02 .27523257E-02 •23470276E-02 .17588353E-02 •10165045E-02 .14464919E-03 ..83644394E-03 ■.19158113E-02 ..30623688E-02 ..42717591E-02 •.73553285E- .12894464E- .67347837E. ..14304087E- -.63364203E- •.78955893E- ..70415776E. -.47728635E- ••18A63294E. .96900150E- .34073779E- .32516051E- •62342092E. .629252A1E- .35o79382E. .37 9 00331E- .14287078E. -.90430397E- -.30308501E. -.98798473E- 04GB 1 03GB 2 04GB 3 04GB * 04GB 5 04GB 6 04GB 7 04GB 8 04GB 9 05GB10 04GB 11 04GB12 04GB13 04GB14 04GB15 04GB16 04GB17 03GB18 04GB19 04GB20 1-18 Table 4.2 Algorithm input for quantile ratio computations for 7 = 20 and selected probabilities 104 .99881322^-01 - . 18871401EC00 -. 26298174E600 .10188944E&00 -.1867o348EcOO -• 223l9488E£00 .10316677ECO0 -. 18321279EC0O -. 19B57915ECO0 .10412579EG00 -• 18398494EC00 -. 18o46 147E tOO •10A90290E£00 -. 18292388EC00 -. 16601550ECOO .10556114EC00 -.18197942EC00 - . 15395222EC00 .106135UE&00 -•18112207EC00 - . 14355844ECO0 .10664597EC00 - . 18O33284EC0O -. 13441 132E COO .1071O766ECO0 -.17939847EC00 - . 12622735EEO0 .10752986EC00 - . 17890961EC00 -. 1 188 1266EC00 .11055327EC00 -. 17350259Et00 -. 67729 170E-01 .11255811EC00 -.16947210EC0O - . 35862646E-0 1 .11411220EC00 -.16611222EC00 -. 12295 1 14E-0 1 .11540613EE00 -. 163 16170EC00 . 65526337E-02 .11652894EC00 -. 160491 17EC00 .11752992EC00 - . 15802537ECOO .11843945EC00 -. 1557 1665EC0O .11927763E&00 . 12005855Et00 . 1261?097E£00 .13087289Et00 .13513473EE00 .139311fl5E£O0 .14367840Ee00 .14856029EC00 ■.15353221E600 ■.15144863E600 .. 13365297EC0O ■•11809116EC00 •.10282062E600 ■•86804248E-01 ..69029590E-01 •.48Q05425E-01 . 15454652E&00 -. 20720647E-01 •1633l914E t o0 .21881934E-01 .16453983E&00 .16587637EC00 .16735834EC00 .16902856Et00 .17o95256E£oO .17323855Et00 . 17608580E&00 •17993202EC00 . 186l3l03E£00 •18703746ECOO •1880407lEtp0 .18916572EC00 . 19044878EC00 •28029184E-01 .34816482E-01 .42409042E-01 .51047592E-01 •61101655E-01 .73184213E-01 •88431180E-01 .10935379EtQ0 .22328486E-01 .35937668E-01 .47921827E-01 .58646772E-01 •68354943E-01 . 13470538EC00 .l744732lEt00 .20155322E&00 .22O03818EC00 .23o994O0EtO0 .23340690EC00 .22289000ECOO . 1828o759E£00 .17509468E&00 . 16608478EC00 .15542836E£O0 .14260015EC00 . 12677782Et00 .10658267E&00 .79411330E-01 .39336341E-01 .14379800Et00 -. 32746960E-01 .14890406E600 -. 44009373E-01 .15437542EC00 -. 56686305E -ol .16O95924EC0O - . 7 1 1 49663E-0 1 •16827035EC00 -. 87967974E-01 '.35992913EC00 . 4l5392l0EC00PB 1 ..2459l4l8Ec00 • 19^3383 IE cOO p B 2 ..18144427EC00 . 8 1915 188E-01 PB 3 ■•13668371EE00 . 14835979E-01 PB 4 ••10290109Ec00 -. 23i 82667E-01PB 5 ■.75935302E-01 - .64539807E-01PB 6 ..53765694E-01 - .79489226E-0lPB 7 ..34582243E-01 - . 94342556E-01 PB 8 ■.18128889E-01 -. 1 183 1225EC00PB 9 ■. 37047 166E -02 -. 130760 10E £00 PB 10 •83634919E-01 -. 167 14673EC00PB1 1 .12771319EC00 -. 17481535EC00PB12 .15459019EE00 -. 14o8 12 16EC00PB13 .17268325EE00 -. 13942427EC00PB 14 .18548354EC00 -. 1 13685 17EC00PB15 .19440329EE00 -. 94569277E-01 PB16 .20074787EC00 -. 66326419E-01PB 17 .20535825EC00 - .4 1Q79595E-01 PBl8 .20822792EC00 -. 283 16703E-01PB19 . 199727B4EC00 . 13768539EC00PB20 •16320573EC00 . 208382 1 1E&00PB2 1 •11620535EC00 . 25985371EC00PB22 .63236123E-01 . 285 18445E C00PB23 .63676307E-02 . 25 194664EC00PB2* ■.53432157E-01 . 203 12472E &00PB25 ■.11362826EC00 . 1 190074 1EC00PB26 -.163 2494 lEeOO -. 23l93232E-01 PB27 ..16586982EC00 -. 3 178 1405E-01PB28 .. 1675 2668EC00 -. 44200475E-01PB29 ..16812189EC0O - . 799 85776E-01PB30 ..1 6635 372 E COO - .957 12586E-01PB31 .••1622 0577EE.00 -.11388992EC00 p B32 ..13361155EE00 -.12405960E£00PB33 ■•1 3762 502 E COO - . 1 3744437 E COOP B3* ..107491 8 2E COO -. l438o85 2ECOOPB35 ..38126106E-01 -.99810101E-01PB36 •.26291367E-01 -.113352 89 EC00PB37 -. 1 193067 IE -01 -.9 29 08 377E-01PB38 •45311859E-02 - . 72725656E-01PB39 .24175104E-01 -.48737463E-01PB40 1-19 Table 4.2 (cont.d) Algorithm input for quantile ratio computations for 7 = 20 and selected probabilities 104 .19194563EC00 • 17684048E£00 -. 10801073E£00 • 48Q35983E-01 -. 25956831E-01PB41 .19374878EC00 . 18722062E600 -. 13273934EG00 . 7864462 lE-01 .19602958EC00 . 20043641E600 -. 164918 19E£00 . 1 1992006EC00 .19916699ECO0 • 21876769E600 -t 2 lo77o93E£00 . 18 1Q7879EC00 .20434456E&00 . 24938679EC0O -. 29026866E 600 . 29253559E COO •.60990423E-02PB42 .61549507E-01PB«3 .l4878758Et00PB44 •34977127EC00PB45 .18999065EE02 ,18976302E£02 ,18954790Et02 ,18934417E£02 ,18915083Et02 ,18896707E£02 18879213EC02 18B62533Et02 ,18846607E£02 1 18831379E£D2 ,18B16802E£02 18802828E602 18789419EC02 18776538EC02 18764l49Et02 ,18752225E£02 ,18740734EC02 ,18729652E£02 ,18718956E C0 2 .18708622EG02 ,18698633E£,02 ,18688967Et02 , 18679609E £ 02 ( 18670541 E t02 ,1866l75oE£02 ,1865322lE£02 ,18644942E £ 02 ,18636 9 00E£02 ,18629o84E£02 ,18621483 E £02 ,166l4o89Eco2 ,1860689lE£02 ,18599883E C o2 ,185 9 3053E£02 .20738106E£01 •19603549EC01 .18532210EC01 .17518204EC01 .16536487EC01 .15642635E601 •14772703EC01 .13943215E601 .13151072E601 .12393493E601 .11667981EC01 .10972293E601 .10304399EC01 .9*t24344E£00 .90447985E600 ,84499013Et0O .78763605Efi0O ,7322 9 297eC00 .67884201E&00 .62717613EC00 •57719394EC00 .52880999EC00 .48l93506EC00 .43649124E600 .39240622EC00 .34 9 6118 9 EC00 •3080458oE£00 ,26765047eC00 •22836946Ee0O .19015123EC00 •l5295027Et00 .1167l922Et00 •81416478E-01 .47002147E-01 -.68135165E-01 -.56305560E-01 -.45935583E-01 -.36R19058E-01 -.28793880E-01 -.21722285E-01 -.15488154E-01 -.99916449E-02 -.51463074E-02 -.87901114E-03 .28743149E-02 .61683989E-02 .90522697E-02 .11568022E-01 .13754334E-01 .15642001E-01 •l726l59iE_oi .18638905E-01 •19796052E-01 .20754952E-01 .21533044E-01 .22146657E-01 .226107UE-01 .22939552E-01 .23143595E-01 ,23?34207e-01 •23221661E-01 ,23U3638e-01 •22920166E-01 .22646488E-01 .22300905E-01 .2I888698E-OI •21413307E-01 . 20886193 E-01 .12961368E-02 .81384021E-03 .44497553E-03 ,1588400«E-03 .60524384E-04 .22491058E-03 .34599073E-03 .43111618E-03 .48834118E-03 .52244011E-03 .53635992E-03 .53802247E-03 •52506773E-03 .50510574E-03 .47689888E-03 .44351639E-03 >40543l23 E -03 .36484835E-03 •3223020*E-03 .27889388E-03 .23499671E-03 ,19137359£-03 .14838210E-03 .10596038E-03 .64496935E-04 .24867369E-04 •13091587E-04 .490U571E-04 .83448234E-04 .11572759E-03 . 14493126E-03 .17364570E-03 . 19939654E-03 .22305861E-03 .23U9701E- .30108261E- -.34198469E- -.61299187E- -.87819327E- -.98675034E- -.85963403E- ..83794802E- ■.71589694E- -.36330222E- -.44050717E- -.26424267E- -.13134717E- -.31182231E- .12343753E- .21282077E- •30406620E. .42741631E- .46914046E. .47332576E- .49439462E. .33347816E- .59856252E. ,34228773 E - .58701783E. .55372522E- .39489766E. .30804721E" •32830352E. .434 9 7 9 67e. .43158790E. .33092842E- •31207246E. ,26 9 15513E- 04GB 1 05GB 2 05GB 3 05GB 4 05GB 5 05GB 6 05GB 7 05GB B 05GB 9 05GB10 05GB11 05GB12 05GB13 06GB14 05GB15 05GB16 05GBl7 05GB1B 05GBl9 05GB20 05GB21 05GB22 05GB23 05GB2* 05GB25 05GB26 05GB27 05GB28 05GB29 05GB30 05GB31 05GB32 05GB33 05GB34 1-20 Table 4.2 (cont'd) Algorithm input for quantile ratio computations for 7 = 20 - 104 and selected probabilities 18586395EC02 18579903EC02 1857357oE£o2 18567387E602 185613S1EC02 18553435EC02 185-+9693EC02 18544061EC02 18538535E602 18533168E£02 18527897E602 18522738EC02 18517685EC02 18512738Efi02 18507891E602 1850?l*lEC02 18498484E&02 18493919Efc02 18489441E602 18483o*7Eco2 18480737EC02 18476506Et02 18472352EC02 18468273E£02 18464266E602 18460331EC02 18456463EC02 18452662EC02 18448925EC02 1844325lE£02 18441638EC02 18A38o85Ec02 18434588E602 1843U48E C 02 18427762E602 18424429Eeo2 18421149E602 18417918EC02 18414736EC02 18411602E£02 •13439743E-01 -.1930607AE-01 -.51268898E-01 -.82479030E-01 -.11296662E&00 -.14276071Efc00 -.17188550EC00 -.20036771EC00 -.22823018E600 -.23549580EC00 -.28218563EC00 -.30831967EC00 -.33391758Ee00 -.35899796EftOO -.38357805Ee00 -.40767434Ee00 -.A3130352EC00 -.45447919EE00 -.A7721687EC00 -.A9953099E600 -.52143367gC00 -.54293791EC00 -.56405332EC00 ..58479864EC00 -.6O517911EC00 -.62520485E600 -.64488938EC00 ..66423982E600 ..68326749E600 -.70l97986Efi00 ..7203872lgt00 ..73849646E600 ..75631660E600 ..77385322E600 -.791H959E600 ..8o8ll624EtOO ..82485279E600 -.84133630EC00 •,85737285Et00 ..87336857E600 •20303866E-01 .19A79166E-01 •19008241E-01 .18298787E-01 .17552489E-01 .16773661E-01 .15964756E-01 .15128473E-01 .14266167E-01 .13380836E-01 .12474427E-01 .11549105E-01 .10605804E-01 .96463758E-02 .86730845E-02 .76866761E-02 .66889109E-02 .56805725E-02 .46613861E-02 •36352490E-02 .26O20571E-02 .15611597E-02 .51487073E-03 -.53639520E-03 -.15916296E-02 -.26B04975E-02 -.37135574E-02 -.47781385E-02 -.58443939E-02 -.69127B01E-02 -.79812405E-02 -•90512067E-02 -.10120109E-01 -.11189053E-01 -.12?57598e-01 -.13325568E-01 -.14392048E-01 -.15456397E-01 -.16320122E-01 -.17581398E-01 •24310394E-03 .26397330E-03 •28165364E-03 .29673328E-03 .30994707E-03 .32107044E-03 .32965613E-03 .33733143E-03 .342767566-03 .34539443E-03 .34778613E-03 .34737733E-03 .34536255E-03 •34190650E-03 .33742661E-03 .33032437E-03 .32281659E-03 .31308538E-03 .30177421E-03 •28834256E-03 .27539960E-03 .26070433E-03 .24464666E-03 .22694679E-03 .2079A633E-03 •18838341E-03 .16768788E-03 .14635781E-03 .12315929E-03 .99637903E-04 .74798507E-04 .49714917E-04 .229U092E-04 ..45763190E-05 ..32291580E-04 ..6U08759E-04 ■.90644380E-04 ■.12111576E-03 ..15173021E-03 ..183399Q8E-03 •21383469E. .20470787E- •92244314E. •10171327E. .77998907E. •18750778E' -.62910041E. -.97494543E. -.10595267E' -.19841424E' -.15192654E- -.29958722E- -.32316795E- -.32446747E- -.38742677E- -.3653560AE. -.A2123373E- -.43891298E- -.41639906E- -.S2258372E- -.A3318215E- ..48378531E- -.A3U0278E- -.45734351E- •.43303713E- ■•34167461E- •.41590888E- ..4O387850E- -.38887090E- ■ •42379284E. ..30348859E- ..26947897E. ■.28i 9 7255e- ..26739235E. -.19806291E- ..18416764E. -.12646606E- •.44705458E- ■. 13418695 E- •.12741381E- ■03CB35 ■05GB36 06CB37 ■05CB38 ■07GB39 ■06GB40 ■06GB41 -06GB42 ■05GB43 •05GB44 •05GB45 ■ 05GB46 ■03GB47 • 05GB48 05GB49 ■ 05GB50 •05GB51 •05GB32 05GB33 05GB54 05GB35 05GB56 03GB37 05GB58 05GB59 05GB60 05GB61 05GB62 05GB63 05G364 05GB*5 05GB66 05GB67 05GB68 05GB69 050B70 05GB71 06GB72 06GB73 06GB74 1-21 Table 4.2 (cont'd) Algorithm input for quantile ratio computations for 7 = 20 and selected probabilities 104 .1840B515EC02 .18405473EC02 .18402476EC02 .18399521EC02 .18396610E602 .18393739E£02 .18390908E602 .ie388116E£,02 .18385363EE02 .1B3826«7EC02 .183799<>9EC02 .88932830EC00 .90485979E600 •92016741E600 .93525639Et00 .95013255EC00 .96480107E600 .97926653EC00 .99353275E600 .10076050EC01 .10214894EC01 .10351866EC01 -.18AA0463E-01 -.19696706E-01 -.20750753E-01 -.21801719E-01 -.22849586E-01 -.23893964E-01 -.24936A49E-01 -.25974433E-01 -.27O08920E-01 -.28039981E-01 -.29O67934E-01 .21609588E-03 .24856010E-03 .28200894E-03 .31613981E-03 .35031696E-03 .38577629E-03 .421U498E-03 .45741560E-03 ,49**7l53E-03 •53056210E-03 56745876E-03 .87816205E- •13526246E. .14553698E- •31536027E- .37884509E- .A0386764E- •52538316E- .60808736E- .6624U90E- .78755335E- .882A92UE- ■06GB75 ■05C876 ■05GB77 05CB78 05GB79 05CB80 05CB81 05GB82 05GB83 05GB84 05GB85 1-22 Table 5. Table of critical values for the Kolmoporov-Smirnov Test. (a) Sample size 25, (b) sample size 30, and (c) asymptotic values All asymptotic values are to be multiplied by (l/N) 3 *. All values adjusted to be no lower than normal where necessary. Distributions Probability Levels (a) .20 .15 .10 .05 .01 No n- parametric No parameters Estimated Exponential distribution Location parameter estimated Shape parameter 1.0 (gamma) Exponential distribution Shape parameter estimated and equal to 1 Scale parameter estimated Gamma distribution Shape parameter estimated and equal to 2 Scale parameter estimated Gamma distribution Shape parameter estimated and equal to 3 Scale parameter estimated Gamma distribution Shape parameter estimated and equal to 4 Scale parameter estimated Gamma distribution Shape parameter estimated and equal to or greater than 8 Scale parameter estimated Gamma distribution Shape parameter known and greater than 3 Scale parameter estimated Normal distribution Location parameter estimated Scale parameter estimated Extreme value distribution Location parameter estimated Scale parameter estimated (a) (b) (c) .208 .190 1.07 .220 .200 1.14 .238 .218 1.22 .264 .242 1.36 .317 .290 1.63 25 30 N (a) b (c) .170 .155 .86 .180 .164 .91 .191 .174 .96 .210 .192 1.06 .247 .226 1.25 25 30 N (a) (b) (c) .165 .152 .84 .173 .159 .89 .185 .169 .95 .204 .184 1.05 .241 .214 1.20 25 30 N (a) (b) (c) .159 .146 .81 .166 .153 .85 .176 .161 .91 .190 .175 .97 .222 .203 1.16 25 30 N (a) (b) (c) .148 .136 .77 .156 .143 .81 .166 .151 .86 .180 .165 .94 .208 .191 1.08 25 30 N (a) (b) (c) .146 .134 .75 .154 .140 .79 .164 .148 .83 .178 .163 .91 .209 .191 1.06 25 30 N (a) (b) (c) .143 .131 .74 .149 .137 .78 .159 .146 .81 * .173 .161* .89 * .203 .186* 1.04 25 30 N (a) (b) (c) .160 .147 .800 .167 .154 .838 .178 .165 .893 .194 .180 .970 .225 .212 1.12 25 30 N (a) (b) (c) .142 .131 .736 .147 .136 .768 .158 .144 .805 .173 .161 .886 .200 .187 1.031 25 30 N (a) (b) (c) .152 .134 .738 .157 .140 .769 .170 .149 .816 .183 .164 .888 .209 .190 1.041 25 30 N ♦adjusted to be no lower than the normal 1-23 Table 6 Plotting positions for ten reliability dewpoints (from Kao, 1968) i 1 2 3 4 5 6 7 8 9 10 x i 2.75 3.1 3.4 3.8 4.1 4.4 4.7 5.1 5.7 6.4 i/(n+l) 9.1 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81.9 91.0 1-24 APPENDIX 2 USE OF ALGORITHM TO DETERMINE QUANTILE RATIO FOR A GIVEN SHAPE PARAMETER AND PROBABILITY The following is an example which shows how to use the algorithm to determine a quantile ratio for a given shape parameter and probability. Given y = 10 and P = .5, the algorithm is f(x,y) = xy +xy +xy +xy +xy . 11 2 2 33 4 4 55 The x. and y. values may be associated with the probability (PB) and 99.99. 25-26 LIMIT LIMIT is the controlling iteration value. If blank or zero, the default value of 10 is chosen. LIMIT is not used in the current version of the Program. 3-6 27-73 AFMT AFMT is the user defined data format. Example of period total, 13 values/card and ICOD = 1. (15, 12, II, 13F4.2) 15 - STN or data set identifier in col. 1-5. 12 - Year of sequence number in col. 6-7. 11 - Card number within sequence # in col. 8. 13F4.2 - Thirteen fields of data with each field 4 cols, in width and an assumed decimal for data recorded to the nearest 0.01. Example for ICOD = 2. (15, 12, 12, 13, 13, F6.2, F6.2, F6.2) 15 - Data set identifier in col. 1-5. 12 - Period number I col. 6-7. 12 - No. of weeks in period J in col. 8-9. 13 - NX = No. of years of nonzero entries, col. 10-12, 13 - NNX = No. of total years, col. 13-15. F6.2 - XBAR = gamma distribution mean, col. 16-21. F6.2 - GAMMA = shape parameter, col. 22-27. F6.2 - BETA = scale parameter, col. 28-33. 74 MILL Input data are not in inches but millimeters (mm) and user wants quantiles converted to mm, code MILL = 1 , otherwise 0. 75-80 BLANK Not used. If the user has multiple data sets to run with uniform characteristics (i.e., all options identical), only one control set of cards is required. Two blank cards terminate the run. 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