Goodness-of-Fit Tests for Regional Generalized Extreme Value

Transcription

1 WATER RESOURCES RESEARCH, VOL. 27, NO. 7, PAGES , JULY 1991 Goodness-of-Fit Tests for Regional Generalized Extreme Value Flood Distributions JAHIR UDDIN CHOWDHURY Institute of Flood Control and Drainage Research, Bangladesh University of Engineering and Technology, Dhaka JERY R. STEDINGER AND LI-HSIUNG LU School of Civil and Environmental Engineering, Cornell University, Ithaca, New York This paper develops critical values and formulas for computing several goodness-of-fit tests for the generalized extreme value (GEV) distribution. These tests can check if data available for a site are consistent with a regional GEV distribution, except for scale, or if the data are consistent with a GEV distribution with a regional value of the shape parameter K. Three tests employ unbiased probabilityweighted moment (PWM) estimators of the L moment coefficient of variation (L-CV), and coefficient of skewness (L-CS) using formulas for their variances in small samples. In a Monte Carlo power study the L-CV test was often more powerful than the Ko!mogorov-Smirnov test at detecting L-C¾ inconsistencies. A test based upon L-CS generally has equal or greater power than the probability plot correlation test at detecting L-CS differences; both are poor at detecting thin-tailed alternatives. Finally, a new chi-square test based upon sample estimates of both the L-CV and L-CS, and their anticipated cross correlation, was much better than other tests at detecting departures from the assumed L-CV, L-CS, or both, particularly when the regional distribution was highly skewed. INTRODUCTION whether a regional GEV distribution is consistent with the available data for a site. A flood frequency relationship for a site where there is Another important but more general issue is how to little or no streamflow record can be constructed by using a categorize the basins in a province into appropriate distinct regional flood frequency model. Use of a generalized ex- "regions" for the identification of a single flood frequency treme value (GEV) distribution as a regional flood frequency distribution for each region, or for development of functions model with an index flood approach has received consider- employing physiographic basin parameters. This is a much able attention. Recent papers discussing the index flood more general issue addressed by Acreman and Sinclair approach include Greis and Wood [1981], Stedinger et al. [1986], Wiltshire [1986a, b, c], Hosking [1987], Burn [1989] [1983], Lettenmaier and Potter [1985], Wallis and Wood and other authors. While the goodness-of-fit tests developed [1985], Hosking et al. [1985b], Hosking and Wallis [1986], here could contribute to such studies, the categorization of Lettenmaier al. [1987], Landwehr et al. [1987], Hosking basins into regions is beyond the scope of this paper. and Wallis [1988], Wallis [1988], Boes et al. [1989], Jin and An important concern is how our goodness-of-fit tests for Stedinger [1989], and Potter and Lettenmaier [1990]. With a a specified regional dimensionless GEV distribution should 6EV index flood procedure the location and shape parame- be used with index flood procedures. If one tested every site ters of a single dimensionless average GEV distribution are in a region with a 1000% test, in a homogeneous region, obtained by pooling information from many sites in a region. 1000% of the sites would on average exhibit a significant lack The implicit assumption for the index flood procedure is that of fit. However, for those sites whose records do not allow floods at every site in the region arise from the same or very rejection of a regional GEV distribution, one would have shnilar distributions, except for scale. Stedinger [1989] suglittle justification for advocating a more complex model. gests the dimensionless regional quantiles might instead be For sites whose records are sufficiently long that they described by a regional physiographic regression relationappear to be significantly at odds with a regional dimensionship; Tasker and Stedinger [ 1986, 1989] develop a weighted less GEV distribution, several actions are reasonable. If the least squares regression model for use in regional analysis which provides an estimate of the standard error of predicphysiographic characteristics of the site are felt to differ from those of other sites in the region, then the statistical and tion which is not inflated by sampling variability. An important question is whether the available flood data physical evidence suggests that the site be assigned to for the site are consistent with a proposed regional GEV anotheregion. Alternatively, if a site does not appear to be distribution for that site. Goodness-of-fit tests can be used to unusual physiographically, and the lack of fit is not highly test whether data at a particular site are consistent with significant, then the lack of fit can be attributed to the a hypothesized regional distribution. Here the Kolmogorovoccasional significant result one should expect if the goodness of fit for many sites were examined. Stairnov test, the probability plot correlation test, and sample L moment ratio tests are investigated for use in testing Finally, one has cases where (1) it is difficulto assign an unusual site with a significant lack of fit to a better region, or Copyright 1991 by the American Geophysical Union. (2) sufficient data are available for a site to clearly demon- Paper number 9!WR strate a significant lack of fit even if the site is not unusual /91/91 WR physiographically. Then it would be reasonable to abandon 1765

2 1766 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION the one-parameter index flood procedure and to use the at-site data to estimate two parameters, corresponding to location and scale, if not also a coefficient of skewness. The coefficient of skewhess, or equivalently the GEV distribution's K, describes the behavior of the upper tail of the distribution and is difficult to estimate reliably with short and moderate length at-site records. However, as sufficient data become available at a site, it is reasonable to progress from one-parameter, to two-parameter, and eventually threeparameter estimation procedures. GEV DISTRIBUTION L-CV= -2 r2 --= [1-2-K]F(I + )/ c ½ + - +,k 3-3 L- CS=,r3=--= A2 I1-2 -K] ' -4 L - kurtosis = r ,2 (5b/ 3 (5c) 1-6(2- ) + 10(3- ) - 5(4- ) where F ( ) is the gamma function [Hosking, 1990]. When K = 0 (Sd) The cumulative distribution function (CDF) of a GEV distribution can be written F(y) = exp {-[1 - K F(y) = {-exp kv 0 < =0 (la) (lb) y = a{ : + [1-(-ln F(y)) ]t c} 0 (2a) y = a{ -In [-In F(y)]} c = 0 (2b) L MOMENTS Probability Weighted Moments and L Moments r2 'k2/a =ln 2/[ + e] (6b) r 3 ---,k 3/A 2 = /' / -2 = where e is Euler's constant equal to With this formulation, a is a scale parameter which can be estimated using the sample mean, s e is a dimensionless L Moment Ratio Estimators location parameter, and is the shape parameter. For K = 0, the GEV distribution reduces to the Gumbe! (EV1) distribu- Let b r be an estimator of/3 r in (4). Then estimates of the tion. The inverses of ( 1 a) and ( 1 b) are L moment ratios r2 and r3 are A promising description of statistical properties of a probability distribution are L moments, developed by Hosking [1990]. L moments are analogous to the conventional product moments but are estimated by linear combinations of order statistics. The analogues of conventional moment ratio estimators, such as the coefficient of variation (CV), coefficient of skewness (CS), and kurtosis are L moment ratio estimators, L-CV, L-CS, and L-kurtosis. L moments estimators are easily constructed using probability-weighted moment (PWM) estimators. The order r PWM can be defined as [Greenwood et al., 1979] fl,. = E{y[F(y)] r } (3) For r = 0, /30 is just the population mean /x. With this definition, (r + 1)/3r is the expected value of the smallest observation in a sample of size (r + 1). L moments can be expressed in terms of linear combinations of PWMs. For the first four L moments [Hosking, 1990]: A3 = 6/32-6B1 + /30 /30 12 = 2B - rio, For a GEV distribution and K =/: 0 (4) A4 = 20/33-30/ fit -/3 0 (5a) q2 -, 2/, 1 = (2bl- bo)/b o '7'3 -= -3/' 2 = (662-6b + bo)/(2b - bo) (6c) (6d) (7a) (7b) Estimators br of PWMs can be obtained using a plotting position to estimate F(yu;) in (3). Hosking et al. [1985a] suggesthe plotting positions (i )/n. Alternatively, an unbiased estimator of/3,. is [Landwehr et al., 1979] 1 (i- 1)(i- 2)...(i - r) b,. =- (n- l)(n- 2)...(n- r)y(i) {81 r/i: 1 where y (o are the ordered values of a sample of size n so that Y(1) <-- ''' <-- Y(i) --<''' <--- Y(n). The unbiased estimators are employed here. Hosking [1986] observes that population L moment ratios are bounded so that I < 1 for r = 3 and 4. For strictly positive random variables, r2 < 1. However, conventional population product moment ratios are not bounded. A disadvantage of the product moment ratios is that when estimated from a finite sample of size n, the sample CV is bounded and cannot exceed (n - 1)0.5, and the sample CS is bounded and cannot exceed (n - 2)/(n - 1) 0'5 [Kirby. 1974]. However, sample L moment ratios -3 and 4 based upon unbiased PWM estimators for samples n >- 4 can take on all feasible values for the population L moment ratios [Hosking, 1986, p. 25], and for strictly positive random variables, r 2 can take on values between zero and one. {In fact, for the sampie y( )... Y(n- ) = 0 and ytn/= 1, 'r= 1.) An important advantage of L moments is that because L moment estimators are linear functions of data, they suffer less from the effects of sampling variability and bias due to squaring and cubing the data [Hosking, 1990]. For the unbiased b r a first-order estimate of the asymp' totic variance of sample L moment ratio estimat0rs is

3 CHOWDHURY ET AL.'- GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION 1767 developed in Appendix A. The solution utilizes the asymptotic distribution of the unbiased estimators b r developed by ttosking et al [1985a ]. Table 1 compares the first-order asymptotic results with the Monte Carlo results based upon 100,000 samples. (The table contains n Var [ '2], n Var [ '3] and n Cov [ '2, '3]-) The simulated variances approach the asymptotic values as the sample size n increases. Approximations of these moments are developed later for use with small n. REGIONAL GEV FLOOD ( UANTILE ESTIMATION PROCEDURES Wallis [1988] proposes a GEV index flood procedure based upon PWM or equivalently L moments. It has four steps: (1) At each site k compute the L moments estimates ' 1 h 2 k, X3. (2) For the region, calculate the average sample size-weighted value of the normalized L moments Xr = nk(xr/x )/(Eff= n ) of order = 2 and 3 across all sites, where nt is the length of record for site k. (3) Using the X, X, and, estimate the parameters and quantiles of the regional dimensionless GEV distribution. (4) A flood quantile for a site is estimated by multiplying the dimensionless regional GEV quantile by an index flood (usually the mean annual flood) for the location, using the available at-site sample or values obtained by a regression on physiographic basin characteristics. Jin and Stedinger [1989] suggest one should not weight by the sample sizes when computing the normalized regional L moments if some sites have much longer records than others, so as not to give them undue influence. However, if some sites have ve short records, some weighting would be advantageous; the optimal weights depend both upon the heterogeneity of the region and the sample sizes [Stedinger and Tasker, 1985]. When concurrent flows are crosscogelated, there are limits on the accuracy with which average regional statistics can be estimated [Stedinger, 1983; Hosking and Wallis, 1988]. Caution should be exercised when using a GEV distribution to model strictly positive random variables such as strea ow and rainfall. For many apparently reasonable combinations of L-CV and L-CS, the probabiliw with which negative values are predicted and would occur in a Monte Carlo simulation is unreasonably large. Thus the GEV distribution is an una eptable probabiliw model for describing strictly positive phenomena with such L-CV/L-CS combinations. Figure 1 shows the probabiliw of negative values with a GEV distribution with different combinations of L-CV and. For the GEV distribution the re lar product moment CV, g/, is about ice the L-CV. The introduction discusses the use of goodness-of-fit tests with the index flood procedure. KOLMOGOROV-SMIRNOV TEST The Kolmogorov-Smirnov (K-S) goodness-of-fit test can be used to check whether observed flood sample at a site is consistent with a regional GEV floodistribution for that site. An advantage of the K-S test is that it givesimulta- neous confidence intervals for all the observations and thus provides a visual goodness-of-fit test. The classical K-S statistics are applicable when the probability distribution completely specified and no parameters are estimated from the observed data. Crutcher [1975] cautions against use of the K-S test with critical values that do not reflect whether or

4 1768 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION, Pr[Y<O] > ' :... '; ',... :... --_ -'",""... -,.1.0.;/o : : -_. "? )o i...!... i!--_l , Fig. 1. The probability assigned to negative values by GEV distributions with different values of K and L-CV, r 2. For strictly positive random variables Y, L-CV/K pairshould not be adopted for which negative values are predicted with a significant probability. not the parameters were estimated. He presents a table of a uniform distribution using subroutine DRNUN in the Inter. critical values of K-S statistics for sample sizes n = 25, 30 national Mathematical Subroutine Library (IMSL). The 10ca. and the asymptotic value when parameters of the exponen- tion parameter s e is obtained from (5b) and (6b) for given r, and tial, gamma, normal, and Gumbel distributions are esti- <. For the selected combinations of r2 and <, the product mated. Stephens [1974] derives formulas for critical values moment CV lies between 0.17 and!.35 (CV is aboutwice as a function of n when parameters of normal and exponen- L-CV). For c between and +0.20, r3 varies from 0.341t0 tial distributions are estimated from a sample. A study by 0.048; the product moment CS decreases from to Spinelli and Stephens [1987] shows that the K-S goodness- The required K-S statistics are calculated as follows. For of-fitest is not as good as three other empirical distribution each sample of GEV random numbers, function (EDF) statistics when location and scale parameters of a two-parameter exponential distribution are estimated. Here the location and shape parameters of the regional GEV D + = max - F(Y i= 1, 2,---, n (11at distribution are known, and only a scale parameter is to be estimated from the data. The K-S test provides bounds within which every obser- D - = max P(y (i)) ' i= 1, 2,---, n (11bl vation should fall if the sample is actually drawn from the hypothesized distribution. Let Yti) be the ordered values of a D=max[D + D-] (11ct sample of size n so that, y( ) _< ß ß <_ Y(i) -<"ß-< Y(n). The test specifies that [Loucks et al., 1981] where the estimate P(Y(i)) of F(y(o ) is obtained by substituting the estimated & for a in (1). Using the mean, 3;, of a generated sample, the estimate & of a is obtained from (10}. Pr F -] - Ko < y(i) < F -1 i- 1 +Ko) = 1-0 This yielded 10 sets of 10,000 values old (denoted by D1, (9) D2, ''', D 0,000) for each n and combination of and K. i=l, 2,---,n Critical values are obtained from the empirical CDF for D: where Pr [ ] denotes probability of the argument inside the K = D 0,0000 brackets, F- denotes the inverse of F ( ), and Ko is the critical value of the test statistic significance level 0. where D 0,0000th denotes the 10,0000th largest D in the To check the consistency of the available flood sample for sequence of 10,000 generated values of D. The reported K-S a site with the regional GEV distribution, F(y) is to be statistics Ks are the average of the 10 values of K. For I0 obtained from the CDF in (1) by using the regional values of < n _< 100 one can employ the approximation and K, and an at-sit estimate of scale. The estimate a of the scale parameter a depends on the mean y of the flood sample and can be obtained from (5a) or (6a) as go (13t & = /{ : + [1 - I'(1 + c 0 (10a) Values of to and 6o 2 are given in Table 2. The accuracy of (13) decreases and it underestimates K o for n < 10. & = /{s e + s} = 0 (10b) Ten sets of 10,000 samples of GEV random variables, each PROBABILITY PLOT CORRELATION TEST of length n = 5, 10, 20, 30, 50, 100 were generated with The probability plot correlation (PPC) was introduced by a = 1.0, r 2 = 0.1(0.1)0.6 and = -0.25(0.05)0.2. This was Filliben [1975] as a test statistic for normality. Looney and accomplished using (2). Values of F(y) were generated from Gulledge [1985] observe that the test compares favorably

5 CHOWDHURY ET AL.: GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION 1769 TABLE 2. Critical Points for the Kolmogorov-Smirnov Test Approximated by (w /n ø'5 - to2/n 2) (Equation(13)) for a Regional GEV Distribution With Estimated Scale Parameter, 10 < n <- 100 Values of co Values of o) 2 Regional c r 2 = 0.1 r2 = 0.3 r 2 = 0.6 r2 = 0.1 r 2 = 0.3 r2 = = o= ! ! = 0.I The r2 are regional values. with tests proposed by Shapiro and Wilk [1965], Shapiro and Francia [1972], and LaBrecque [1977]. Vogel [1986] derives PPC test statistics for the Gumbel distribution. Vogel and Kroll [1989] consider the Weibull and uniform distributions. The PPC test is a measure of linearity of a probability plot. If the sample to be tested is actually drawn from the hypothesized distribution, the plot of the ordered observations y(i) versus corresponding expected values M i for the hypothesized distribution is expected to be nearly linear and the correlation coefficient will be near to one. The PPC coefficient can be obtained from [Filliben, 1975] 5 (Y(i)- 37)(Mi- M) - i=1 P = 0.5 (14) I ( Y (i) - 37) 2 (Mi - ) 2] i--1 i=1 where 37 = Y = 1 yi/n and ]fir = Z?=l Mi/n. The value of p will be unaffected by either a scaling or a translation. Therefore it tests whether the shape parameter (K or r3) of the at-site distribution equals that o[ a hypothesized GEV distribution since values of a regional location parameter : and the at-site scale parameter c would not influence p. The Mi can be obtained from the GEV inverse function (2) by using an estimate P(Y(i)) of F(y(i)) along with the regional value of c. A plotting position formula is needed f0r thestimator P(Y(i)). After Cunnane [1978], we employ P(Y(o) = (i- 0.4)/(n + 0.2) (15) For each of the previously generated samples of GEV mnd0rn variables the corresponding values of m i are obtained from the GEV inverse function (2) by using plotting position formula (15) and the correct regional values of and c. The correlation coefficient, p., for a sample is obtained using (14). Critical values for the PPC coefficient are obtained from the empirical CDF for 9 as described earlier for the Kolmogorov-Smirnov test statistic. The critical points of the distribution of p are near to unity. Thus it is convenient to tabulate critical points of (1 - p) instead. The critical points do not depend upon the L-CV of a hypothesized GEV distribution, only on c and n. The results are summarized in Table 3 for various c of the GEV distribution. The 1% points for tc < actually increase with n for n -> 30. One rejects the hypothesized GEV distribution if the observed value of p is smaller than the critical value. TEST BASED UPON L-CV The distribution of 9 r is asymptotically normal [Hosking, 1990]. It is assumed here that the 92 and the 93 obtained from finite sample sizes are also normally distributed. Hosking [1990] shows by simulation that the normal approximation for the distribution of the 93 is very good when samples are drawn from a Gumbel distribution. The normal approximation permits development of a simple test of whether 92 and 93 for a site are significantly different from '2 and r3 of the regional GEV. Approximations for the Var [92] for small n can be obtained by correcting the asymptotic variance. The Monte Carlo analysis employed the 10 sets of 10,000 samples of GEV random variables with c = 1.0, r (0.1)0.6 and tc = -0.25(0.05)0.20. For each sample, an estimate of?2 was obtained by using (7a) and (8). The variance of the 100,000 values of 92 was determined for each combination of regional r2 and to. An approximation of the variance of 92 is Var [cr 2] 2,n = (v + v2/n)/n (16) Table 4 gives values of, and '2 for various r2 and tc of the regional GEV distribution. When r2 is small, ½2,n is close to Var [92] obtained from the asymptotic n Var [?2]-

6 1770 CHOWDHURY ET AL.: GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION TABLE 3. Values of 1000 (1 - Pc) Where Pc is the Critical Value for the Probability Plot Correlation Test n t =-0.30 t =-0.20 t =-0.10 c = 0.00 = = = O= = i The hypothesis of a GEV distribution with specified K is rejected if the observed value of the probability plot correlation coefficient is less than the critical value. Note that 1% points (0 = 0.01) for < actually increase with n for n > 30. By comparing the statistic O2,n' -0 5 (?2 - r2) with the different from the r3 for a regional G EV distribution. The B critical values of a standard normal distribution, one can test of the regional GEV distribution are known from (5c)and whether the?2 at a site is significantly different from the r2 (6c) for the regional value of K. for the regional GEV distribution. A similar test statistic is proposed by Hosking et al. [1985a] for determining whether the shape parameter K is TEST BASED UPON L-CS zero when fitting a GEV distribution. That test, based on br calculated using the plotting positions (i )/n, com- The approximations for Vat [ ] for finite n were also pares the statistic k(n/0.5633) ø'5 with the critical values 0fa determined by using Monte Carlo simulation to correct the standard normal distribution. Hosking et al. [1985a] demonfirst-order approximate asymptotic Vat [?3]- The asymptotic strate that the test is almost as powerful as the modified Var [?3] is independent of r2. The variance of?3 as a function likelihood ratio test which was found by Hosking [1984] t0be of n is approximately the best test of the hypothesis c = 0. Var [?3]- 03,n = ( ql + rl2/n)/n (17) for values of r/i and */2 given in Table 5. TEST BASED UPON L-CV AND L-CS By comparing the statistic O,n ø'5 (?3-?3) with the Based on the two sample L moment ratios?2 and?., a critical values of a standard normal distribution, one can test single goodness-of-fit test can be developed to check whether the 73 at a site, based on (7b) and (8), is significantly whether assumed values of r 2 and % for a regional GEV TABLE 4. Coefficients for the Estimate (v + v2/n)/n of the Variance of Sample L-CV (Equation (16)) When Population is GEV Truest r 2 =0.1 Values of v Distribution With Specified r2 and 0.2 r 2 = 0.3 r 2 = 0.4 r2 =0.5 r2 =0.6 r2 =0.1 Values of v2 r2 = 0.2 r2 = 0.3 r2 = 0.4 r2 =0.5 z2 =0' ! O i The r2 are true values.

7 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION 1771 TABLE 5. Coefficients for the Estimate ( /1 + 2/n)/n of the Variance of Sample L-CS (Equation (17)) When Population is GEV Distribution for Specified K or Equivalent '3 True K True ' distribution equal the values for the population from which the sample was drawn. It is importanthat '2 and '3 are correlated, and the correlation is larger with a thicker-tailed GEV distribution ( c < 0) and small r 2. The asymptotic correlation, corr [ '2, 73], can be estimated ascov [ 2, '3]/[( Vat [ r2]) ø'5 (Vat[ ])0. ] using the formulas in Appendix A. However, a Monte Carlo experiment again provides approximations in Table 6 for the finite-sample c0 elation co [ 2, 3]' co [q2, 3] p = I + 2/n (18) For thin-tailed distributions, the finite-sample correlation c0e cients are close to the asymptotic values. Letting = o test statistic becomes 2,n( 2-2) (19a) standard chi-square distribution with 2 degrees of freedom, one can test whether either or both of '2 and '3 are significantly different from the r2 and r 3 for the regional GEV distribution. Appendix B provides an analytical approximation for the type II error of this test against GEV alternatives. POWER COMPARISONS A Monte Carlo study evaluated the power of the five goodness-of-fit tests studied using 100,000 samples of length n = 20 or 50, from several alternative distributions. For these tests, the null hypothesis was that generated observations were drawn from a GEV distribution with assumed regional values of r 2 and r 3. The type I error, or significance level, was set at 0 = 5% so the null hypothesis would be rejected 5% of the time were it true. The power of the test is the frequency with which the null hypothesis was actually rejected when samples were drawn from the specified true distribution or alternative distribution. Table 7 summarizes the power (in percent) for eight GEV alternatives, where true L-CV and/or L-CS are different from those of the assumed regional GEV distribution. There are three sets of cases: (1) true r 2 regional r2 but true K = regional K; (2) true r 2 regional r2 and true K - regional K; and (3) true v2 = regional r2but true K - regional K. In cases when the assumed and true L-CV and L-CS were the same, tests based upon 72 and '3 did have the specified 5% type I error, indicating that the normality assumption was satisfactory. Table 8 summarizes the power of the tests against 12 non-gev alternatives so that the true population is a twoparameter exponential, normal, or gamma distribution. There are two sets of cases: r2 of true population is equal to r2 of the G EV distribution; and '2 of true population is not equal to r 2 of the G EV distribution. In all cases, the scale parameter was unity. The L moment ratios of exponential, normal, and gamma distributions are given by Hosking [1990]. The shape parameters of the gamma distributions were 7.70, 3.28, and 1.72 when r2 was 0.2, 0.3, and 0.4, respectively. 2 = 8 re - a (20) The L-CV tests using O2- ø'5 ( '2- r2) and the K-S test are Monte Carlo experiments confirmed that X 2 has approxi- able to detect lack of consistency of at-site L-CV with that of mately a chi-square distribution with 2 degrees of freedom the hypothesized regional GEV distribution. The K-S test is f0rn 20. not very sensitive to differences in K (or L-CS) ß The 2,t ' -0 5 By comparing the statistic X 2 with the critical values of a ( '2 - r2) test has more power than the K-S test when L-CV TABLE 6. Coefficients for the Estimate (œ + c02/n)/n of the Sample Correlation Coefficient of L-CV and L-CS (Equation (18)) When Population is GEV Distribution With Specified '2 and K Values of œ 1 Values of co 2 True r2 = 0.1 r2 = 0.2 ' = 0.3 r 2 = 0.4 '2 = 0.5 r2 = 0.6 '2 = 0.1 r 2 = 0.2 r2 = 0.3 r2 = 0.4 '2 = 0.5 '2 = I The r 2 are true values.

8 1772 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION TABLE 7. Observed Power (in Percent) of Goodness-of-Fit Tests at 5% Significance Level When the True Population Belongs to GEV Family, but Parameters Are Different From That of the Assumed Regional GEV Distribution True GEV Assumed GEV Power for n = 20 Power for n = 50 z2,c r2,c K-S Test r 2 Test X 2 Test r3 Test PPC Test K-S Test '2 Test X 2 Test '3 Test PPCTes Different r2 but Same K " " '-' " '-' Different and Different Same r2 but Different '" "' ! '" "' '" "' '" "' '" '" '" "' '" "' '" "' '" "' '" '" Three dots denote values which were found to be the anticipated 5% because true population has same *2 as hypothesizedistribution in case of '2 test and true population has same as hypothesized distribution in case of '3 and PPC tests. of the true GEV is larger than that of the hypothesized GEV. Results are similar when the true population did not belong to the GEV family. The L-CS tests using ½,.5 (.3- r3) and the PPC have similar power. They do reasonably well when the true population has a thicker tail than that of the hypothesized regional GEV distribution corresponding to a true K less than the postulated,c. Filliben [1975] shows that the PPC was particularly powerful at rejecting thick-tailed alternatives, and weak at rejecting thin-tailed alternatives. The results in Tables 7 and 8 are consistent with that observation. The composite X 2 test using '2 and '3 turns out in many cases to be better than either the corresponding '2 or '3 test when p? is large, corresponding to K -< -0.1 and small '2- This occurs because the correlation between '2 and '3 allows for a more powerful test of whether the sample estimates of the L-CV and L-CS are jointly consistent with the hypothesized G EV distribution. Viewed another way, the values of tr 2 and '3 help reduce the effective variation in the other when p? is large. Figure 2 shows how for ( = -0.2, sample estimates of r2 and are clustered in narrow and parallel ellipses in the r2-r3 space and how the variability in '2 is smaller with smaller '2. For,c = -0.2 and '2 = 0.2, the test is very sensitive to departures in either r 2 or '3, or both. Like the L-CV test which it incorporates, it is more powerful in cases when the assumed L~CV is smaller than the true value for the population from which the samples are drax 'n. The X 2 test's power was well approximated by the analytical approximation in Appendix B. A REGIONAL HOMOGENEITY TEST Based upon sample L moment ratios, a statistican be developed to test whether the distributions of the observations at all sites in a region are equal to the hypothesized regional distribution, except for scale. Thus one could construct a test of whether the L moment ratios (r2, r3) or r2, r3, '4) for every site are the same. if the data support rejection of the hypothesis, further investigation might support subdivision of the region, or the use of somexplanatory variables [see Tasker and Stedinger, 1986]. However, the power0f such tests depends both on the actual heterogeneity 0fa region and on the number of sites available to testhe hypothesis of homogeneity [Hosking, 1987]. Assume that data from K sites are available, and let ) equal the sample rth L moment ratio for site k, k = 1,'", K. For a region, the sample size-weighted estimate of the mean of the -p is ',q = Z nkq' nk (21} k=! 1

9 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION 1773 TABLE 8. Observed Power (in Percent) of Goodness-of-Fit Tests at 5% Significance Level When the True Population is Exponential, True Population Assumed Regional GEV Type r2 r2 K Normal or Gamma Power for n = 20 Power for n = 50 K-S '2 x 2 '3 PPC K-S?2 X 2 '3 PPC Test Test Test Test Test Test Test Test Test Test Same '2 two-parameter exponential I normal gamma Different r2 two-parameter exponential ! normal gamma Let X 2 defined (20) for site k be denoted X' k). Then a composite test for all sites in the region could employ K (22) The statistic X' R) has approximately a standard chi-square shape parameter ( c or r3) of the at-site data is different frqm distribution with 2(K - 1) degrees of freedom if the that of the regional GEV distribution. For GEV alternatives observations available at each site are independent. Ideally, their performance is about the same, though both are weak {2!) could be replaced by the mean vector that minimizes at detecting thin-tailed alternatives. For non-gev alterna- (22), especially when 99 is large. CONCLUSION from that of the regional GEV distribution, though the L-CV test is more powerful when the L-CV of the regional model is less than the L-CV for the actual data. However, an important advantage of the K-S test is that it gives simultaneous confidence bounds for all the observations, and thus can provide a visual goodness-of-fit criterion. The PPC test and the L-CS test consider whether the tives, the L-CS test generally has greater power than the PPC test. A chi-square test based upon sampl estimates of both the L-CV and L-CS, and their anticipated cross correlation, is When a GEV distribution serves as a regional model, the relatively good at detecting departures from the regional consistency of floodat available at a site with the regional L-CV, L-CS, or both. It is much better than other tests when GEV model can be tested using several goodness-of-fit the postulated regional distribution highly skewed with a statistics. The K-S test and the L-CV test are good for thick tail (K-< -0.1). Like the L-CV test, it does well when checking whether the L-CV of the at-site data is different the assumed L-CV is smaller than the true value.

10 1774 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION o o :3 0.2 O0 0 0 O0 0.0 ß o o o o * x 2 = 0.20 o 2 = I I I I I I Fig. 2. One hundred pairs of '2 and?3 were calculated from randomly generated samples of size 20 from GEV distributions with < = (corresponding to *3 = 0.31), and r 2 of either 0.20 or The plotted points illustrate how the correlation between the two sample L moment ratios makes it possible to distinguish between the two populations when '2 and r3 are considered simultaneously, even though *3 is the same for the two populations. APPENDIX A: ASYMPTOTIC VARIANCE OF SAMPLE L MOMENT RATIOS Expanding the estimators ( 2/.:) and (, 3/, 2) of 'r 2 and r3 in Taylor series about (A2/A ) and (X3/A2), and retaining first-order terms, Var [ '2], Var [%3] and Coy [ '2,%3] are to first order Var [rr2] = Var [i 2/i ] - X½ Var IX!] + Var [X2] - 2 X} Cov [ 1, 23 (23a) Var [q3] = Var [i 3/i 2] 4 Var [ 2] +,k"-j Var [, 3] A Cov[X2, i31 Cov [ 2, 3] =Cov IX 2/i, i 3/i 2] A3 =x x 2 cov [i, 2] 1 x Coy [, i3] 1 XlX2 Cov [i2, i3] (23b) X 1. 2 Var [i 2] (23c) The L moments Xl, X2 and X 3 are given by (4). The variances and covariances of L moment estimators i, 2 and 3 are elements of the matrix [AVA T] where A is the coefficient matrix of the PWM vectors [/30,/31,/32] r in (4) and V is the matrix of asymptotic variances of unbiased PWM estimators, b0, b l, and b2 in (8). The matrix A is A (24) Following Hosking et al. [1985a] and Hosking [1986], elements Vi, j of the symmetrical matrix V for the firsthree unbiased PWM estimators are 2-2 : o 2 V2, 2 = 2 2 V1,1 -n <2[r(l + 2 <)- r2(1 + t<)] (25a) IlK 3-2 : O 2 V3, [r(1 + 2K)G(«)- r2(1 + K)] (250) 2 IF(1 + 2 )G( )- 1'2(1 + )] (25c) 2 V!,2 = V2,1 2n < 2 [2-2 r( ) + (1-2]- K)F2(1 + )] (25d1

11 CHOWDHURY ET AL.' GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION a + V2,3 = V3, 2 = 2n*: (2-2 _ )F2(1 + :)] Vi,3 = V3,1-2n*: 2 [ 3-2 F(1 + 2*:)- 2r(1 + (25e) + 2( K)F2(! + *:)] (25f) Here G( ) is a special case of the hypergeometric function, and can be calculated as [Oberhettinger, 1964]: or, equivalently, G(x) = 2Fi(*:, 2,:; 1 + *:; -x) 2*: 2 o F(2*: + m) (-x) G(x) =! + F( 1 + 2*:) m =l *: (26a) + m m! (26b) APPENDIX B: POWER OF THE X 2 STATISTIC TEST---AN ANALYTIC APPROACH This appendix derives an analytic approximation for the power of the X 2 test described in the text, equation (20). The ; 2 statistic can be written as 9 - r 2 Var ( r2) f3 r3 L COv (2/'2, 93) or in a matrix notation: Coy ( r2, or3) Var (rr 3) '3 'r (27) X 2 = [t - ix]rs-i[t - ix] (28) where t = ( r 2, '3)r Ix = (r2 r3)r and S is the covariance matrix of ( '2, '3). In moderate to large samples, the L moment ratios are approximately normally distributed, so t has approximately a multivariate normal distribution N(ix, S). For normally distributed t, a X 2 test with the null hypothesis Ho' X = IXo, S = S o versus the alternative hypothesis Ha' p. = Ixa, S = Sa, has power r= Pr {It- ix0] TS -I[t - ixo] 2 > X'(2, - 0) t --- N(ixa, Sa)} (29) The distribution of the variable X equal to [t - o] rsd-1 It- I%] in (29) is not a noncentral chi-square distribution if So is not the covariance of t, as is assumed here under the alternative hypothesis. However, E(X) and Var (X) are [see Seber, 1977, pp. 13 and 41] E(X) = Tr [S - Sa] + [ixa- Ix0]rS - [Ix, - ix0] (30) Var (X)= 2Tr [S 'ISaS -ISa] Monte Carlo simulations show that the distribution of X = [t - [t0] rsd-1 [t - 0] was well approximated by a gamma distribution with the corresponding mean and variance. Using (29)-(30) to estimate rr for the cases reported in Table 7, the differences between analytic approximations of r and the simulated values are generally less than 3 percentage points, and never exceed 5 percentage points. Acknowledgments. The research work was carried out while J.U.C. was a visiting Fulbright Scholar from the Institute of Flood Control and Drainage Research, Bangladesh University of Engi- neering and Technology. Support was also provided by the National Science Foundation grant CME J.R.M. Hosking's cooperation and assistance is gratefully acknowledged. REFERENCES Acreman, M. C., and C. D. Sinclair, Classification of drainage basins according to their physical characteristics: An application for flood frequency analysis in Scotland, J. Hydrol., 84, , Boes, D.C., J. H. Heo, and J. D. Salas, Regional flood quantile estimation for a Weibull model, Water Resour. Res., 25(5), , Burn, D. H., Cluster analysis as applied to regional flood frequency, J. Water Resour. Plann. Manage. Div. Arn. Soc. Civ. Eng., 115(5), , Crutcher, H.-L., A note on the possible misuse of the Kolmogorov- Smirnov test, J. Appl. Meteorol., 4(8), , Cunnane, C., Unbiased plotting positionstoa review, J. HydroI., 37, , Filliben, J. J., The probability plot correlation coefficient test for normality, Technometrics, 17(1), 11!-117, Greenwood, J. A., J. M. Landwehr, N. C. Matalas, and J. R. Wallis, Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resour. Res., 15(5), , Greis, R. M., and E. F. Wood, Regional flood frequency estimation and network design, Water Resour. Res., 7(4), , (Correction, Water Resour. Res., 9(2), , Hosking, J. R. M., Testing whether the shape parameter is zero in the generalized extreme value distribution, Biometrica, 71, , Hosking, J. R. M., The theory of probability weighted moments, Res. Rep. RC12210, IBM Res., Yorktown Heights, N.Y., Hosking, J. R. M., Regional homogeneity, Review of Statistical Flood Frequency Estimation, Open File Rep. 6, Inst. of Hydrol., Wallingford, England, Hosking, J. R. M., L-moments: Analysis and estimation of distributions using linear combinations of order statistics, J. R. Star. Soc., Set. B, 52(1), , Hosking, J. R. M., and J. R. Wallis, The value of historical data in flood frequency analysis, Water Resour. Res., 22(11), , Hosking, J. R. M., and J. R. Wallis, The effect of intersite dependence on regional flood frequency analysis, Water Resour. Res., 24(4), , Hosking, J. R. M., J. R. Wallis, and E. F. Wood, Estimation of the generalized extreme value distribution by the method of probability weighted moments, Technometrics, 27(3), , 1985a. Hosking, J. R. M., J. R. Wallis, and E. F. Wood, An appraisal of the regional flood frequency procedure in the UK Flood Studies Report, Hydrol. Sci. J., 30(1), , 1985b. Jin, M., and J. R. Stedinger, Flood frequency analysis with regional and historical information, Water Resour. Res., 25(5), , Kirby, W., Algebraic boundedness of sample statistics, Water + 4{[ a- Ix0]rS - SaS - [Ix a - Ix0] ) (31) Resour. Res., 0(2), , Since X is in a quadratic form, it must be nonnegative. A LaBrecque, J., Goodness-of-fit tests based on non-linearity gamma distribution with mean and variance given by (30) probability plots, Technometrics, 19, , Landwehr, J. M., N. C. Matalas, and J. R. Wallis, Probability and (31)is the natural generalization of the chi-square weighted moments compared with some traditional techniques in distribution to use to approximate the distribution of X for estimating Gumbel parameters and quantiles, Water Resour. approximating -in (29). Res., 15(5), , 1979.

12 1776 CHOWDHURY ET AL.: GOODNESS OF FIT FOR GENERALIZED EXTREME VALUE DISTRIBUTION Stedinger, J. R., K. Potter, D. Kibler, and G. Tasker, Comments on Landwehr, J. M., G. D. Tasker, and R. D. Jarrett, Discussion on "Relative accuracy of log Pearson III procedures" by J. R. Wallis "Regional flood frequency estimation and network design" by and E. F. Wood, J. Hydraul. Eng., Ili(7), , N. P. Greis and E. F. Wood, Water Resour. Res., 19(5), Lettenmaier, D. P., and K. W. Potter, Testing flood frequency 1345, estimation methods using a regional flood generation model, Stephens, M. A., EDF statistics for goodness of fit and SOme Water Resour. Res., 21(12), , comparisons, J. Am. Stat. Assoc., Theory Methods Sect. Lettenmaier, D. P., J. R. Wallis, and E. F. Wood, Effect of regional 69(347), , ' heterogeneity on flood frequency estimation, Water Resour. Res., Tasker, G. D., and J. R. Stedinger, Regional ske with weighted L$ 23(2), , regression, J. Water Resour. P!ann. Manage. Div. Am. Soc. Cir. Looney, S. W., and T. R. Gulledge, Jr., Use of the correlation Eng., 112(2), , coefficient with normal probability plots, Am. Stat., 39(1), 75-79, Tasker, G. D., and J. R. Stedinger, An operational GLS model for hydrologic regression, J. Hydrol., 111 (1--4), , Loucks, D. P., J. R. Stedinger, and D. A. Haith, Water Resource Vogel, R. M., The probability plot correlation coefficient test f0rthe Systems Planning and Analysis, pp and , chap. normal, lognorma!, and Gumbel distributional hypothesis, Water 3, Prentice-Hall, Englewood Cliffs, N.J., Resout'. Res., 22(4), , Oberhettinger, F., Hypergeometric functions, in Handbook of Vogel, R. M., and C. N. Kroll, Low-flow frequency analysis using Mathematical Functions, Appl. Math. Ser., vol. 55, edited by M. probability-plot correlation coefficients, J. Water Resour. Plann. Abramowitz and I. A. Stegun, chap. 15, National Institute of Manage. Div. Am. Soc. Civ. Eng., 115(3), , Standards and Technology, Gaithersburg, Md., Wallis, J. R., Catastrophes, computing and containment: Living in Potter, K. W., and D. P. Lettenmaier, A comparison of regional our restless habitat, Speculations Sci. Technol., 11(4), , flood frequency estimation methods using a resampling method, Water Resour. Res., 26(3), , Wallis, J. R., and E. F. Wood, Relative accuracy of log Pearson IIi Seber, G. A. F., Linear Regression Analysis, John Wiley, New procedures, J. Hydraul. Eng., 111(7), , York, Wiltshire, S. E., Regional flood frequency analysis, Shapiro, S.S., and R. S. Francia, An approximate analysis of statistics, Hydro!. $ci. J., 31(3), , 1986a. variance test for normality, J. Am. Stat. Assoc., 67, , Wiltshire, S. E., Regional flood frequency analysis, II, Multivariate classification of drainage basins Britain, Hydrol. Sci. J., 31(3}, Shapiro, S.S., and M. B. Wilk, An analysis of variance test for , 1986b. normality (complete samples), Biometrica, 52, , Wiltshire, S. E., Identification of homogeneous regions for flood SpineIll, J. J., and M. A. Stephens, Tests for exponentiality when frequency analysis, J. Hydrot., 84, , 1986c. origin and scale parameters are unknown, Technometrics, 29(4), , J. U. Chowdhury, Institute of Flood Control and Drainage Re- Stedinger, J. R., Estimating a regional flood frequency distribution, search, Bangladesh University of Engineering and Technology, Water Resour. Res., 19(2), , Dhaka- I000, Bangladesh. Stedinger, J. R., Using historical and regional information flood L.-H. I,u and J. R. Stedinger, School of Civil and Environmental frequency analyses, paper presented at Pacific International Sem- Engineering, Cornell University, Ithaca, NY inar on Water Resources Systems, Jpn. Soc. of Hydrol. and Water Resour., Tomamu, Japan, Aug. 8-10, Stedinger, J. R., and G. D. Tasker, Regional hydrologic analysis, 1, (Received January 8, 1990; Ordinary, weighted and generalized least squares compared, revised December 20, 1990; Water Resour. Res., 21(9), , accepted January 4, 1991.) I, Homogeneity