Detailed abstract of the manuscript: Hosking, J.R.M., J.R. Wallis, and E.F. Wood. (1985). Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments. Technometrics 27(3), 251--261.
Abstract
Hosking et al. (1985) use the method of probability-weighted moments,
introduced by Greenwood et al. (1979), to estimate the parameters of the
generalized extreme value distribution (GEVD) with parameters
location=\(\eta\), scale=\(\theta\), and
shape=\(\kappa\). Hosking et al. (1985) derive the asymptotic
distributions of the probability-weighted moment estimators (PWME), and compare
the asymptotic and small-sample statistical properties (via computer simulation)
of the PWME with maximum likelihood estimators (MLE) and Jenkinson's (1969)
method of sextiles estimators (JSE). They also compare the statistical
properties of quantile estimators (which are based on the distribution parameter
estimators). Finally, they derive a test of the null hypothesis that the
shape parameter is zero, and assess its performance via computer simulation.
Hosking et al. (1985) note that when \(\kappa \le -1\), the moments and probability-weighted moments of the GEVD do not exist. They also note that in practice the shape parameter usually lies between -1/2 and 1/2.
Hosking et al. (1985) found that the asymptotic efficiency of the PWME (the limit as the sample size approaches infinity of the ratio of the variance of the MLE divided by the variance of the PWME) tends to 0 as the shape parameter approaches 1/2 or -1/2. For values of \(\kappa\) within the range \([-0.2, 0.2]\), however, the efficiency of the estimator of location is close to 100 are greater than 70 Hosking et al. (1985) found that the asymptotic efficiency of the PWME is poor for \(\kappa\) outside the range \([-0.2, 0.2]\).
For the small sample results, Hosking et al. (1985) considered several possible forms of the PWME (see equations (8)-(10) below). The best overall results were given by the plotting-position PWME defined by equations (9) and (10) with \(a=0.35\) and \(b=0\).
Small sample results for estimating the parameters show that for \(n \ge 50\) all three methods give almost identical results. For \(n < 50\) the results for the different estimators are a bit different, but not dramatically so. The MLE tends to be slightly less biased than the other two methods. For estimating the shape parameter, the MLE has a slightly larger standard deviation, and the PWME has consistently the smallest standard deviation.
Small sample results for estimating large quantiles show that for \(n \ge 100\) all three methods are comparable. For \(n < 100\) the PWME and JSE are comparable and in general have much smaller standard deviations than the MLE. All three methods are very inaccurate for estimating large quantiles in small samples, especially when \(\kappa < 0\).
Hosking et al. (1985) derive a test of the null hypothesis \(H_0: \kappa=0\)
based on the PWME of \(\kappa\). The test is performed by computing the
statistic:
$$z = \frac{\hat{\kappa_{pwme}}}{\sqrt{0.5663/n}} \;\;\;\; (1)$$
and comparing \(z\) to a standard normal distribution (see
zTestGevdShape). Based on computer simulations using the
plotting-position PWME, they found that a sample size of \(n \ge 25\) ensures
an adequate normal approximation. They also found this test has power comparable
to the modified likelihood-ratio test, which was found by Hosking (1984) to be
the best overall test of \(H_0: \kappa=0\) of the thirteen tests he considered.
More Details
Probability-Weighted Moments and Parameters of the GEVD The definition of a probability-weighted moment, introduced by Greenwood et al. (1979), is as follows. Let \(X\) denote a random variable with cdf \(F\), and let \(x(p)\) denote the \(p\)'th quantile of the distribution. Then the \(ijk\)'th probability-weighted moment is given by: $$M(i, j, k) = E[X^i F^j (1 - F)^k] = \int^1_0 [x(F)]^i F^j (1 - F)^k \, dF \;\;\;\; (2)$$ where \(i\), \(j\), and \(k\) are real numbers.
Hosking et al. (1985) set
$$\beta_j = M(i, j, 0) \;\;\;\; (3)$$
and Greenwood et al. (1979) show that
$$\beta_j = \frac{1}{j+1} E[X_{j+1:j+1}] \;\;\;\; (4)$$
where
$$E[X_{j+1:j+1}]$$
denotes the expected value of the \(j+1\)'th order statistic (i.e., the maximum)
in a sample of size \(j+1\). Hosking et al. (1985) show that if \(X\) has a
GEVD with parameters location=\(\eta\), scale=\(\theta\), and
shape=\(\kappa\), where \(\kappa \ne 0\), then
$$\beta_j = \frac{1}{j+1} \{\eta + \frac{\theta [1 - (j+1)^{-\kappa} \Gamma(1+\kappa)]}{\kappa} \} \;\;\;\; (5)$$
for \(\kappa > -1\), where \(\Gamma()\) denotes the
gamma function. Thus,
$$\beta_0 = \eta + \frac{\theta [1 - \Gamma(1+\kappa)]}{\kappa} \;\;\;\; (6)$$
$$2\beta_1 - \beta_0 = \frac{\theta [\Gamma(1+\kappa)] (1 - 2^{-\kappa})}{\kappa} \;\;\;\; (7)$$
$$\frac{3\beta_2 - \beta_0}{2\beta_1 - \beta_0} = \frac{1 - 3^{-\kappa}}{1 - 2^{-kappa}} \;\;\;\; (8)$$
Estimating Distribution Parameters Using the results of Landwehr et al. (1979), Hosking et al. (1985) show that given a random sample of \(n\) values from some arbitrary distribution, an unbiased, distribution-free, and parameter-free estimator of the probability-weighted moment \(\beta_j = M(i, j, 0)\) defined above is given by: $$b_j = \frac{1}{n} \sum^n_{i=j+1} x_{i,n} \frac{{i-1 \choose j}}{{n-1 \choose j}} \;\;\;\; (9)$$ where the quantity \(x_{i,n}\) denotes the \(i\)'th order statistic in the random sample of size \(n\). Hosking et al. (1985) note that this estimator is closely related to U-statistics (Hoeffding, 1948; Lehmann, 1975, pp. 362-371).
An alternative “plotting position” estimator is given by: $$\hat{\beta}_j[p_{i,n}] = \frac{1}{n} \sum^n_{i=1} p^j_{i,n} x_{i,n} \;\;\;\; (10)$$ where $$p_{i,n} = \hat{F}(x_{i,n}) \;\;\;\; (11)$$ denotes the plotting position of the \(i\)'th order statistic in the random sample of size \(n\), that is, a distribution-free estimate of the cdf of \(X\) evaluated at the \(i\)'th order statistic. Typically, plotting positions have the form: $$p_{i,n} = \frac{i-a}{n+b} \;\;\;\; (12)$$ where \(b > -a > -1\). For this form of plotting position, the plotting-position estimators in (10) are asymptotically equivalent to the U-statistic estimators in (9).
Although the unbiased and plotting position estimators are asymptotically equivalent (Hosking, 1990), Hosking and Wallis (1995) recommend using the unbiased estimator for almost all applications because of its superior performance in small and moderate samples.
Using equations (6)-(8) above, i.e., the three equations involving \(\beta_0\), \(\beta_1\), and \(\beta_2\), Hosking et al. (1985) define the probability-weighted moment estimators of \(\eta\), \(\theta\), and \(\kappa\) as the solutions to these three simultaneous equations, with the values of the probability-weighted moments replaced by their estimated values (using either the unbiased or plotting posistion estiamtors in (9) and (10) above). Hosking et al. (1985) note that the third equation (equation (8)) must be solved iteratively for the PWME of \(\kappa\). Using the unbiased estimators of the PWMEs to solve for \(\kappa\), the PWMEs of \(\eta\) and \(\theta\) are given by: $$\hat{\eta}_{pwme} = b_0 + \frac{\hat{\theta}_{pwme} [\Gamma(1 + \hat{\kappa}_{pwme}) - 1]}{\hat{\kappa}_{pwme}} \;\;\;\; (13)$$ $$\hat{\theta}_{pwme} = \frac{(2b_1 - b_0)\hat{\kappa}_{pwme}}{\Gamma(1 + \hat{\kappa}_{pwme}) (1 - 2^{-\hat{\kappa}_{pwme}})} \;\;\;\; (14)$$ Hosking et al. (1985) show that when the unbiased estimates of the PWMEs are used to estimate the probability-weighted moments, the estimates of \(\theta\) and \(\kappa\) satisfy the feasibility criteria $$\hat{\theta}_{pwme} > 0; \, \hat{\kappa}_{pwme} > -1$$ almost surely.
Hosking et al. (1985) show that the asymptotic distribution of the PWME is multivariate normal with mean equal to \((\eta, \theta, \kappa)\), and they derive the formula for the asymptotic variance-covariance matrix as: $$V_{\hat{\eta}, \hat{\theta}, \hat{\kappa}} = \frac{1}{n} G V_{\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2} G^T \;\;\;\; (15)$$ where $$V_{\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2}$$ denotes the variance-covariance matrix of the estimators of the probability-weighted moments defined in either equation (9) or (10) above (recall that these two estimators are asymptotically equivalent), and the matrix \(G\) is defined by: $$G_{i1} = \frac{\partial \eta}{\partial \beta_{i-1}}, \, G_{i2} = \frac{\partial \theta}{\partial \beta_{i-1}}, \, G_{i3} = \frac{\partial \kappa}{\partial \beta_{i-1}} \;\;\;\; (16)$$ for \(i = 1, 2, 3\). Hosking et al. (1985) provide formulas for the matrix $$V_{\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2}$$ in Appendix C of their manuscript. Note that there is a typographical error in equation (C.11) (Jon Hosking, personal communication, 1996). In the second line of this equation, the quantity \(-(r+s)^{-k}\) should be replaced with \(-(r+s)^{-2k}\).
The matrix \(G\) in equation (16) is not easily computed. Its inverse, however, is easy to compute and then can be inverted numerically (Jon Hosking, 1996, personal communication). The inverse of \(G\) is given by: $$G^{-1}_{i1} = \frac{\partial \beta_{i-1}{\partial \eta}}, \, G^{-1}_{i2} = \frac{\partial \beta_{i-1}{\partial \theta}}, \, G^{-1}_{i3} = \frac{\partial \beta_{i-1}{\partial \kappa}} \;\;\;\; (17)$$ and by equation (5) above it can be shown that: $$\frac{\partial \beta_j}{\partial \eta} = \frac{1}{j+1} \;\;\;\; (18)$$ $$\frac{\partial \beta_j}{\partial \theta} =\frac{1 - (j+1)^{-\kappa}\Gamma(1+\kappa)}{(j+1)\kappa} \;\;\;\; (19)$$ $$\frac{\partial \beta_j}{\partial \kappa} = \frac{\theta}{j+1} \{ \frac{(j+1)^{-\kappa}[log(j+1)\Gamma(1+\kappa)-\Gamma^{'}(1+\kappa)]}{\kappa} - \frac{1 - (j+1)^{-\kappa}\Gamma(1+\kappa)}{\kappa^2} \} \;\;\;\; (20)$$ for \(i = 1, 2, 3\).
Estimating Distribution Quantiles
If \(X\) has a GEVD with parameters location=\(\eta\),
scale=\(\theta\), and shape=\(\kappa\), where \(\kappa \ne 0\),
then the \(p\)'th quantile of the distribution is given by:
$$x(p) = \eta + \frac{\theta \{1 - [-log(p)]^{\kappa} \}}{\kappa} \;\;\;\; (21)$$
\((0 \le p \le 1)\). Given estimated values of the location, scale, and shape
parameters, the \(p\)'th quantile of the distribution is estimated as:
$$\hat{x}(p) = \hat{\eta} + \frac{\hat{\theta} \{1 - [-log(p)]^{\hat{\kappa}} \}}{\hat{\kappa}} \;\;\;\; (22)$$
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Greenwood, J.A., J.M. Landwehr, N.C. Matalas, and J.R. Wallis. (1979). Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressible in Inverse Form. Water Resources Research 15(5), 1049--1054.
Hoeffding, W. (1948). A Class of Statistics with Asymptotically Normal Distribution. Annals of Mathematical Statistics 19, 293--325.
Hosking, J.R.M. (1985). Algorithm AS 215: Maximum-Likelihood Estimation of the Parameters of the Generalized Extreme-Value Distribution. Applied Statistics 34(3), 301--310.
Hosking, J.R.M. (1990). \(L\)-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society, Series B 52(1), 105--124.
Hosking, J.R.M., and J.R. Wallis (1995). A Comparison of Unbiased and Plotting-Position Estimators of \(L\) Moments. Water Resources Research 31(8), 2019--2025.
Jenkinson, A.F. (1969). Statistics of Extremes. Technical Note 98, World Meteorological Office, Geneva.
Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, pp.4-8.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.
Lehmann, E.L. (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, Oakland, CA, 457pp.