Goodness-of-fit tests for the generalized pareto distribution.

Goodness-of-fit tests for the generalized pareto distribution. Abstract: Tests of fit are given for the generalized Pareto distribution (GPD) based on Cramer-von Mises statistics. Examples are given to illustrate the estimation techniqu es and the goodness-of-fit procedures. The tests are applied to the exceedances over g iven thresholds for 238 river flows in Canada; in general, the GPD provides an adequat e fit. The tests are useful in deciding the threshold in such applications; this method is investigated and also the closeness of the GPD to some other distributions that might be used for long-tailed data.KEY WORDS: Exceedances; Extreme values; Hydrology; Threshold selection.The generalized Pareto distribution (GPD) has the following distribution function:F(x) = 1 - [(1 - kx/a).sup.1/k], (1)where a is a positive scale parameter and k is a shape parameter. The density function isf(x) = (1/a)[(1 - kx/a).sup.(1-k)/k]; (2)the range of x is 0 [less than or equal to] x < [infinity] for k [less than or equal to] 0 and 0 [less than or equal to] x [less than or equal to] a/k for k > 0. The mean and variance are [mu] = a/(l + k) and [[sigma].sup.2] = [a.sup.2]/{[(1+k).sup.2](1+2k)}; thus the variance of the GPD is finite only for k > -0.5. For the special values k = 0 and 1, the GPD becomes the exponential and uniform distributions, respectively. The n ame generalized Pareto was given by Pickands (1975); the distribution is sometimes ca lled simply Pareto when k < 0. In this case, the GPD has a long tail to the right and h as been used to model datasets that exhibit this form in several areas of appliedIn particular, the GPD is used to model extreme values. This application was discussed by many authors--for example, Hosking and Wallis (1987), Smith (1984, 1989, 1990), Davison (1984), and Davison and Smith (1990). Smith (1990) gave an excellent revie w of the two most widely used methods in this field, based on generalized extreme value distributions and on the GPD. In hydrology, the GPD is often called the "peaks over thresholds" (POT) model since it is used to model exceedances over threshold levels i n flood control. Davison and Smith (1990) discussed this application in their section 9, using river-flow exceedances for a particular river over a period of 35 years. The autho rs fit the GPD for exceedances over a series of thresholds and also calculated the Kolm ogorov-Smirnov and Anderson-Darling statistics to test fit. In the article, they mentione d the lack of tests for the GPD [echoing a remark made earlier by Smith (1984)]. In t he absence of such tests, they used tables for testing exponentiality, which, as the aut hors pointed out, give too high critical values. This dataset is tested for GPD in Section 3.In this article, goodness-of-fit tests are given for the GPD, based on the Cramer-von M ises statistic [W.sup.2] and the Anderson-Darling [A.sup.2]. We concentrate on the mo st practical case in which the parameters are not known. Estimation of parameters is d iscussed in Section 1, and the goodness-of-fit tests are given in Section 2. In Section 3, they are applied to exceedances for a Canadian river, and it is shown how the tests can be used to help select the threshold in the POT model, following ideas suggested by Davison and Smith (1990). Moreover, the Davison-Smith example is revisited, using the GPD test and the Anderson-Darling statistic. The tests are then used to model the exceedances over thresholds of 238 Canadian river-flow series; they indicate the adeq uacy of the GPD fit. The technique of choosing the threshold is based on a stability pro perty of the GPD; the efficacy of the technique when the true exceedance distribution i s not GPD is examined in Section 4. In Section 5, we investigate the versat ility of the GPD as a distribution that might be used for many types of data with a long tail and no mode in the density. Finally, the asymptotic theory of the tests is given in Section 6.1. ESTIMATION OF PARAMETERSBefore the test of fit can be made, unknown parameters in (1) must first be estimated. The asymptotic theory of the test statistics requires an efficient method of estimation;we shall use maximum likelihood. Although it is theoretically possible to have datas ets for which no solution exists to the likelihood equations, in practice this appears to be extremely rare. For example, in our examination of the 238 Canadian rivers, maxim um likelihood estimates of the parameters could be found in every case. Thus we shall assume that such estimates exist for the dataset under test. Hosking and Wallis (198 7), Castillo and Hadi (1997), and Dupuis and Tsao (1998) studied other methods of est imating the parameters. One of these methods is the use of probability-weighted m oments; although there appear to be some advantages in terms of bias, Chen and Bal akrishnan (1995) and Dupuis (1996) later showed that this technique is not always fea sible.We now discuss estimation by maximum likelihood. Suppose that [x.sub.1],..., [x.sub. n] is a given random sample from the GPD given in (1), and let [x.sub.(1)] [less than or equal to] [x.sub.(2)] [less than or equal to] ... [less than or equal to] [x.sub.(n)] b e the order statistics. We consider three distinct cases--Case 1, in which the shape par ameter k is known and the scale parameter a is unknown; Case 2, in which the shape parameter k is unknown, and the scale parameter is known; Case 3, in which both pa rameters a and k are unknown. Case 3 is the most likely situation to arise in practice. The log-likelihood is given byL(a, k) = -n log a - (1 - 1/k) x [summation over (i=1] [n] log(1 - [kx.sub.i]/a) for k [not equal to] 0 = - n log a - [summation over (i=1] [n] [x.sub.i]/a for k = 0. (3) The range for a is a > 0 for k [less than or equal to] 0 and a > [kx.sub.(n)] for k > 0. When k < 1/2, Smith (1984) showed that, under certain regularity conditions, the maximum likelihood estimators are asymptotically normal and asymptotically efficient. When 0.5 [less than or equal to] k < 1, Smith (1984) identified the problem as nonre gular, which alters the rate of convergence of the maximum likelihood estimators. For k [greater than or equal to] 1, and as n [right arrow] [infinity], the probability approac hes 1 that the likelihood has no local maximum. We now consider Cases 1 to 3 separa tely.Case 1 (k known, a unknown).For this case, we have the following result. Proposition 1. For any known k with k < 1, the maximum likelihood estimate of a exist s and is unique.Proof. For k = 0 (the exponential distribution), the result is well known. Suppose that k [not equal to] 0; then a, the maximum likelihood estimate of a, will be a solution a of [partial]L(a, k)/[partial]a = 0, which may be simplified to [L.sub.1](a) = 0, where [L. sub.1](a) = n - (1 - k) [summation over (i=1/n)] [x.sub.i]/(a - k[x.sub.i]). The value of [[partial].sup.2]L(a, k)/[partial][a.sup.2] at a = a is -(1 - k) [summation over (i=1/ n)] [x.sub.i] ([a - k[x.sub.i]).sup.-2]/a < 0, which implies that at a = a, the likelihood function attains its maximum value. Moreover, the function [L.sub.1] (a) is an increasi ng function on the range of a because [partial][L.sub.1] (a)/[partial]a = (1 - k) [summ ation over (i=1/n)] [x.sub.i]/ [(a - [kx.sub.i]).sup.2] > 0; the function can take negati ve and positive values; thus it cuts the a axis exactly at one point. Hence, a is unique.Case 2 (a known, k unknown).In this situation, there may or may not exist a maxi mum likelihood estimate for k. To see this, consider the likelihood function L(a, k) give n in (3) for -[infinity] < k [less than or equal to] a/[x.sub.(n)]. Since a is known, the l ikelihood will be regarded as a function L(k) of k only. Set [k.sup.*] = a/[x.sub.(n)]; t hen[lim.sub.k[right arrow][infinity]] L(k) = - [infinity], (4)[lim.sub.k[right arrow][k.sup.*]] L(k) = - [infinity] if [k.sup.*] < 1, (5)and[lim.sub.k[right arrow][k.sup.*]] L(k) = + [infinity] if [k.sup.*] > 1. (6)Note that the value of k is not restricted to be less than or equal to 1. From (4) and(5), it follows that, if [k.sup.*] < 1, there is at least one maximum. For a fixed sample of size n, Pr([k.sup.*] < 1) = 1 - [[1 - [(1 - k).sup.1/k]].sup.n] > 1 - [(3/4).sup.n].Similarly, if [k.sup.*] > 1, it follows from (4) and (6) that there may or may not exis t a local maximum.Case 3 (both parameters unknown).Maximum likelihood estimation of the paramet ers a and k when both are unknown was discussed in detail by Grimshaw (1993). This case is similar to Case 2: In principle, maximum likelihood estimates for a and k may not exist. However, as stated previously, this is unlikely in practical applications, espe cially when the sample size is reasonably large. To find a solution, Davison (1984) poin ted out that, by a change of parameters to [theta] = k/a and k = k, the problem is re duced to a unidimensional search; we search for [theta], which gives a local maximum of the profile log-likelihood (the log-likelihood maximized over k). This is[L.sup.*]([theta]) = - n - [summation over (n/i=1)] log(1 - [theta][x.sub.i]) - n log [ - [(n[[theta].sup.-1] [summation over (n/i=1)] log(1 - [theta][x.sub.i])] (7)for [theta] < 1/[x.sub.(n)]. Suppose that a local maximum [theta] of (7) can be found; thenk = -([n.sup.-1]) [sumaration over (i=1] .sup.n] log(1 - [theta][x.sub.i]) (8)anda = k/[theta]. (9)2. GOODNESS-OF-FIT TESTSIn this section, the Cramer-von Mises statistic [W.sup.2] and the Anderson-Darling stati stic [A.sup.2] are described. The Anderson-Darling statistic is a modification of the Cra mer-von Mises statistic giving more weight to observations in the tail of the distribution, which is useful in detecting outliers. The null hypothesis is [H.sub.0]: the random sam ple [x.sub.1],..., [x.sub.n] comes from Distribution (1).When parameters a and k are known in (1), the GPD is completely specified; we call t his situation Case 0. Then the transformation [z.sub.i] = F([x.sub.i]) produces a z sam ple that will be uniformly distributed between 0 and 1 under [H.sub.0]. Many tests of the uniform distribution exist, including Cramer-von Mises tests (see Stephens 1986), so we shall not consider this case further. In the more common Cases 1, 2, and 3, when one or both parameters must be estimated, the goodness-of-fit test procedure is as fo llows:1. Find the estimates of unknown parameters as described previously, and make the tr ansformation [z.sub.(i)] = F([x.sub.(i)]), for i = 1,..., n, using the estimates where nec essary.2. Calculate statistics [W.sup.2] and [A.sup.2] as follows:[W.sup.2] = [summation over (i=1/n)] [{[z.sub.(i)] - (2i - 1)/(2n)}.sup.2] + 1/(12n) and[A.sup.2] = - n - (1/n) [summation over (i=1/n)] (2i - 1)[log{[z.sub.(i)]} + log{1 - [z. sub.(n+1-i)]}].Tables 1 and 2 give upper-tail asymptotic percentage points for the statistics [W.sup.2] and [A.sup.2], for values of k between -0.9 and 0.5, and for Cases 1, 2, and 3. In C ases 2 and 3, where k must be estimated, the appropriate table should be entered at k, and in all tables, if k or k is greater than 0.5, the table should be entered at k = 0.5. Critical points for other values of k can be obtained by interpolation; linear interpola tion in Table 2 for [A.sup.2], Case 3, gives a maximum error in [alpha] from 0.0011 t o 0.0003 as [alpha] moves through the important range R; 0.10 [greater than or equal to] [alpha] [greater than or equal to] 0.005 and is less than 0.003 for the two values S; [alpha] = 0.5 and 0.25; for [W.sup.2] the corresponding figures are 0.0025 to 0.0 004 over R and less than 0.007 for S. For the much less important cases 1 and 2, line ar interpolation in Table 1 gives maximum error for [A.sup.2], Case 1, from 0.009 to 0. 0017 over R, and 0.014 for S; for [W.sup.2], Case 1, the figures are 0.0055 t o 0.001 5 over R and 0.008 for S. For Case 2, the maximum errors are smaller by a factor of approximately 10.The points in these tables were calculated from the asymptotic theory to be presented in Section 6. Points for finite n were generated from Monte Carlo samples, using 10,000 samples for each combination of n and k. The results show that these points converge quickly to the asymptotic points for all three cases, a feature of statistics [W.sup.2] and [A.sup.2] found also in many other applications. The asymptotic points can then b e used with good accuracy for, say, n [greater than or equal to] 25. More extensive ve rsions of Tables 1 and 2, and also tables showing the convergence to the asymptotic p oints, were given by Choulakian and Stephens (2000).3 ?In this section we consider two specific examples and then discuss the overall results when the GPD is applied to 238 Canadian rivers.In the first example, the data are n = 72 exceedances of flood peaks (in [m.sup.3]/s) of the Wheaton River near Carcross in Yukon Territory, Canada. The initial threshold va lue [tau] is 27.50 (how this was found will be explained later), and the 72 exceedance s, for the years 1958 to 1984, rounded to one decimal place, were 1.7, 2.2, 14.4, 1.1, 0.4, 20.6, 5.3, 0.7, 1.9, 13, 12, 9.3, 1.4, 18.7, 8.5, 25.5, 11.6, 14.1, 22.1, 1.1, 2.5, 14.4, 1.7, 37.6, 0.6, 2.2, 39, 0.3, 15, 11, 7.3, 22.9, 1.7, 0.1, 1.1, 0.6, 9, 1.7, 7, 20.1, 0.4, 2.8, 14.1, 9.9, 10.4, 10.7, 30, 3.6, 5.6, 30.8, 13.3, 4.2, 25.5, 3.4, 11.9, 21.5, 2 7.6, 36.4, 2.7, 64, 1.5, 2.5, 27.4, 1, 27.1, 20.2, 16.8, 5.3, 9.7, 27.5, 2.5, 27. The ma ximum likelihood estimates of the parameters are k = -0.006 and a = 12.14. We now apply the tests presented in Section 2; the values of the test statistics are [W.sup.2] = 0.2389 (p value < 0.025) and [A.sup.2] = 1.452 (p value < 0.010), so the GPD do es not fit the dataset well at the threshold value of 27.50. Next we follow the techniqu e of Davison and Smith (1990), who suggested raising the threshold until the GPD fits the new values of the remaining exceedances. Here the threshold was raised successiv ely by the value of the smallest order statistic, which is then deleted, until the p value s of the [A.sup.2] and [W.sup.2] statistics exceeded 10%. This happens when six order statistics are deleted; the threshold value is then 27.50 + 0.60 = 28.10. The GPD now fits the data quite well: Details are given in Table 3. This example shows how the te sts may help in choosing a threshold value in the POT model.In this connection, we next revisit the example given by Davison and Smith (1990). Th eir tables 5 and 6 give the parameter estimates (Case 3) and the values of the Anders on-Darling statistic for a range of thresholds from 140 down to 70 at intervals of 10 u nits. The values of the Anderson-Darling statistic were compared with 1.74, the 5% poi nt for a test for exponentiality since no test for GPD was available, and the statistics w ere not nearly significant by this criterion. Now Table 2 can be used to give the asymp totic 5% critical values [z.sub.5] for [A.sup.2] corresponding to the estimate of the par ameter k. The first two and last two threshold results are given in Table 4. Davison an d Smith pointed out the sudden increase in the value of [A.sup.2] at threshold level 70; against the exponential point this value is still not significant, but using Table 2, it fall s just at the critical 5% level.The Wheaton River in the previous example is one of 238 Canadian rivers for which we have similar data. We now examine how well the GPD fits the exceedances for the ot her rivers. First one must decide the threshold level for a given river flow. The first thr eshold estimate, [tau], was chosen so that the number of exceedances per year could be modeled by a Poisson distribution; see, for instance, Todorovic (1979). This was do ne by taking [tau] such that, if N[tau] is the number of exceedances, the mean of N[t au] divided by its variance was approximately 1. The Poisson assumption will be tested more rigorously later. After the threshold level was chosen, the maximum likelihood e stimates of the parameters k and a were calculated and the [W.sup.2] and [A.sup.2] t ests applied. Table 5 gives the frequencies of p values for the 238 rivers. It is clear th at the GPD fits quite well; using [W.sup.2], only 9 gave p values less than 0.01, and u sing [A.sup.2], there were only 15. At the 0.05 level, these figures are 34 using [W.su p.2] and 49 using [A.sup.2]. The results demonstrate also that [A.sup.2] is more sensi tive (and therefore more powerful) than [W.sup.2] against possible outliers in the tail, as was suggested in Section 2. More details were given by Choulakian and Stephens (2000). For the 49 "rejections" using [A.sup.2], the threshold was increased, as described in the first example, by deleting the smallest order statistics until the p value became larger than 0.10. Then only 10 of the 49 sets were still rejected as GPD by [A.sup.2]. Finally, for these 10 rejected river flows, the Poisson assumption was tested by the Cramervon Mises tests given by Spinelli and Stephens (1997). Only one dataset reject ed the Poisson assumption.Another result of interest is that 229 of the 238 values of k were between -0.5 and 0. 4; this confirms the findings of Hosking and Wallis (1987), who restricted attention to -0.5 < k < 0.5, where the great majority of values fall.4. POWER OF THE FAILURE-TO-REJECT METHODThe method of choosing the threshold by using as many exceedances as possible subje ct to passing a test for GPD can be investigated for power as follows. The data analyst wishes to fit a GPD to the exceedances far enough in the tail. She\he therefore starts with as many as possible, say 100, and tests; if the test fails, she/he omits the small est values one by one and tests again until the test yields acceptance for the GPD. Th e efficiency of this procedure can now be investigated when the true distribution is not GPD. Suppose, for instance, the true exceedance distribution is Weibull with parameter 0.75, and suppose, starting with sample size 100, the sample fails the GPD test until K lower-order statistics have been removed; then it passes. Statistic K will vary from s ample to sample, but its mean will give a measure of the power of the test. The stand ard deviation and other statistics from the distribution of K are also of interest. This ha s been investigated for various alternative distributions for the t rue exceedances; Tabl e 6 provides these results. They are based on 1,000 Monte Carlo samples of initial size n = 100, 50, or 30. Only statistic [A.sup.2] is reported because [A.sup.2] outperforms [W.sup.2] at detecting tail discrepancies. Column 3 gives the initial rejection rate for a GPD test with parameters estimated, at test size 0.05; this is also the power in a con ventional study of power against the specified alternative. Subsequent columns give the mean, standard deviation, the three quartiles, and the maximum values for K. Thus for the Weibull(0.75) alternative, 509 of the 1,000 samples of size 100 were rejected as GPD initially; of these rejected samples, the mean number of order statistics to be dele ted before acceptance was 8.169 with standard deviation 9.178. The distribution of K is long-tailed; the quartiles are 2, 5, and 11, respectively, but one sample needed 52 or der statistics removed to achieve acceptance by the GPD test.In Table 6, the distributions were chosen to give a reasonable starting rejection rate. S everal other distributions could be supposed as alternatives (e.g., the half-normal, half-Cauchy, or gamma distributions), but these can be made very close to the GPD by suit able choice of parameters, so the initial power is small, and very few low-order statisti cs, if any, need be deleted to achieve GPD acceptance (cf. the numerical results in Sec.5). In their section 9, Davison and Smith suggested that the exceedances for their ex ample might possibly be truly modeled by a mixture of two populations, and in Table 6 we have included three mixtures of GPD as possible true models. The test statistic dis tinguishes quite effectively between these mixtures and a single GPD.For all alternative distributions considered in Table 6, it may be seen that, as n increas es, the mean, standard deviation, and maximum of K all increase; the results also sho w the increase in the power of the test as the sample size increases.5. VERSATILITY OF THE GPDAs was suggested previously, some distributions often fitted to long-tailed distributions may be brought close to a GPD by suitable choice of k and a. This closeness was inves tigated, and results are given in Table 7. In this table, for example, the standard half-normal and standard half-Cauchy distributions (respectively, the distribution of [absolut e value of X] when X has the standard normal or the standard Cauchy distribution) are compared with the GPD(0.293,1.022) for the first and GPD(-0.710, 1.152) for the sec ond; the standard lognormal is compared with GPD(-0.140, -1.406). The choice of the GPD parameters can be made by equating the first two moments of the GPD and those of the distribution compared, but this does not produce as good a fit as the following maximum likelihood procedure.A sample of 500 values was taken from the half-normal distribution, and a GPD fit was made, estimating the parameters by maximum likelihood, first solving (7) for [theta], and then obtaining k, a from (8) and (9). This was repeated 100 times, and the avera ge values of the k, a values were used as the parameters in the GPD. Although the m atches are clearly not perfect, the error in [alpha], the probability level of a given perc entile, is small when one distribution is used instead of another. Similar comparisons ar e given for several Weibull distributions, and a gamma distribution. The exponential (W eibull with parameter 1) is omitted because it is a special case of the GPD with k = 0. Again the two distributions are quite close for the Weibull parameter greater than 1 (where the Weibull has a mode), but the match is less good for Weibull with paramete r less than 1--for example, the Weibull(0.5), or the Weibull(0.75)--where the density ri ses to infinity at X = 0.Overall, the results suggest that a GPD could often be used as a model for data with a long tail when neither a mode nor an infinite density is suggested by the nature of the variables or by the data themselves.6. ASYMPTOTIC THEORY OF THE TESTSIn this section we summarize the asymptotic theory of Cramer--von Mises tests. The c alculation of asymptotic distributions of the statistics follows a procedure described, for instance, by Stephens (1976). It is based on the fact that [y.sub.n] (z) = [square root of (term)]n{[F.sub.n](z) - z}, 0 [less than or equal to] z [less than or equal to] 1, w here [F.sub.n](z) is the empirical distribution function of the z set, it tends to a Gaussi an process y(z) as n [right arrow] [infinity] and the statistics are functionals of this pr ocess. The mean of y(z) is 0: We need the covariance function [rho](s, t) = E{y(s)y (t)}, 0 [less than or equal to] s, t [less than or equal to] 1. When all the parameters are known, this covariance is [[rho].sub.0](s, t) =min(s, t) - st. When parameters areestimated, the covariance will depend in general on the true values of the estimated p arameters. However, if the method of estimation is efficient, the covariance will not de pend on the scale parameter a but will depend on the shape parameter k. We illustrat e for Case 3 only. As the sample size n [right arrow] [infinity], the maximum likelihoo d estimators (a, k) have a bivariate normal distribution with mean (a, k) and variance-covariance matrix [SIGMA], where[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)When both parameters a and k are estimated, the covariance function of y(z) becomes[[rho].sub.3](s, t) = [[rho].sub.0](s, t) - {g(s)}'[summation over]g(t), (11)where s = F(x), and g(s) = ([g.sub.1](s), [g.sub.2](s))' is a vector having coordinates [g.sub.1] (s) = [partial]F / [partial]a = (1 - s) {1 - [(1 - s).sup.-k]}/(ak)and[g.sub.2] (s) = [partial]F / [partial]k = (1 - s)k log(1 - s) - 1 + [(1 - s).sup.-k]/[k.su p.2].When [SIGMA] and g(s) are inserted into (11), [[rho].sub.3](s, t) will be independent of a. When k [greater than or equal to] 0.5, the maximum likelihood estimates of a an d k are superefficient in the sense of Darling (1955), and then the covariance and the resulting asymptotic distributions will be the same as for k = 0.5. Thus, if 0.5 [less th an or equal to] k [less than or equal to] 1, the table should be entered at k = 0.5, as described in Section 2. In Case 1 the covariance of y(z) becomes[[rho].sub.1](s, t) = [[rho].sub.0](s, t) - (1 - 2k)[g.sub.1] (s)[g.sub.1] (t) (12)and for Case 2 it becomes[[rho].sub.2](s, t) = [[rho].sub.0](s, t) - (1 - k)(1 - 2k)[g.sub.2] (s)[g.sub.2] (t)/2. (1 3)In both these cases, at k = 0.5, the asymptotic covariance becomes [[rho].sub.0](s, t). This is the same as for a test for Case 0, when both a and k are known, and the asy mptotic points are the same as for such a test (see Stephens 1986).The Cramer-von Mises statistic [W.sup.2] is based directly on the process y(z), while [A.sup.2] is based on the process w(z) = y(z)/[{z(1 - z)}.sup.1/2]; asymptotically the distributions of [W.sup.2] and [A.sup.2] are those of [W.sup.2] = [[integral].sup.1.sub. 0] [y.sup.2](z)dz and [A.sup.2] = [[integral].sup.1.sub.0] [w.sup.2](z)dz. The asymptot ic distributions of both statistics are a sum of weighted independent [[chi square].sub.1] variables; the weights for [W.sup.2] must be found from the eigenvalues of an integra l equation with the appropriate [[rho].sub.j](s, t) for Case j as kernel. For [A.sup.2], [[rho].sub.A](s, t), the covariance of the w(z) process, is [[rho].sub.j](s, t)/[{st(1 - s) (1 - t)}.sup.1/2], and this is the kernel of the integral equation. Once the weights are known, the percentage points of the distributions can be calculated by Imhof's method. For details of these procedures, see, for example, Stephens (1976). ACKNOWLEDGMENTSWe thank the editor and two referees for their many helpful suggestions. This work wa s supported by the Natural Science and Engineering Council of Canada. REFERENCESCastillo, E., and Hadi, A. S. (1997), "Fitting the Generalized Pareto Distribution to Data, " Journal of the American Statistical Association, 92, 1609-1620.Chen, G., and Balakrishnan, N. 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