Detailed abstract of the manuscript:
Castillo, E., and A. Hadi. (1994). Parameter and Quantile Estimation for the Generalized Extreme-Value Distribution. Environmetrics 5, 417--432.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Abstract
Castillo and Hadi (1994) introduce a new way to estimate the parameters and
quantiles of the generalized extreme value distribution (GEVD)
with parameters location=\(\eta\), scale=\(\theta\), and
shape=\(\kappa\). The estimator is based on a two-stage procedure using
order statistics, denoted here by “TSOE”, which stands for
two-stage order-statistics estimator. Castillo and Hadi (1994) compare the TSOE
to the maximum likelihood estimator (MLE; Jenkinson, 1969; Prescott and Walden, 1983)
and probability-weighted moments estimator (PWME;
Hosking et al., 1985).
Castillo and Hadi (1994) note that for some samples the likelihood may not have a local maximum, and also when \(\kappa > 1\) the likelihood can be made infinite so the MLE does not exist. They also note, as do Hosking et al., 1985), that when \(\kappa \le -1\), the moments and probability-weighed moments of the GEVD do not exist, hence neither does the PWME. (Hosking et al., however, claim that in practice the shape parameter usually lies between -1/2 and 1/2.) On the other hand, the TSOE exists for all values of \(\kappa\).
Based on computer simulations, Castillo and Hadi (1994) found that the
performance (bias and root mean squared error) of the TSOE is comparable to the
PWME for values of \(\kappa\) in the range \(-1/2 \le \kappa \le 1/2\).
They also found that the TSOE is superior to the PWME for large values of
\(\kappa\). Their results, however, are based on using the PWME computed
using the approximation given in equation (14) of Hosking et al. (1985, p.253).
The true PWME is computed using equation (12) of Hosking et al. (1985, p.253).
Hosking et al. (1985) introduced the approximation as a matter of computational
convenience, and noted that it is valid in the range \(-1/2 \le \kappa \le 1/2\).
If Castillo and Hadi (1994) had used the true PWME for values of \(\kappa\)
larger than 1/2, they probably would have gotten very different results for the
PWME. (Note: the function egevd with method="pwme" uses
the exact equation (12) of Hosking et al. (1985), not the approximation (14)).
Castillo and Hadi (1994) suggest using the bootstrap or jackknife to obtain
variance estimates and confidence intervals for the distribution parameters
based on the TSOE.
More Details
Let \(\underline{x} = (x_1, x_2, \ldots, x_n)\) be a vector of
\(n\) observations from a generalized extreme value distribution with
parameters location=\(\eta\), scale=\(\theta\), and
shape=\(\kappa\) with cumulative distribution function \(F\).
Also, let \(x(1), x(2), \ldots, x(n)\) denote the ordered values of
\(\underline{x}\).
First Stage
Castillo and Hadi (1994) propose as initial estimates of the distribution
parameters the solutions to the following set of simultaneous equations based
on just three observations from the total sample of size \(n\):
$$F[x(1); \eta, \theta, \kappa] = p_{1,n}$$
$$F[x(j); \eta, \theta, \kappa] = p_{j,n}$$
$$F[x(n); \eta, \theta, \kappa] = p_{n,n} \;\;\;\; (1)$$
where \(2 \le j \le n-1\), and
$$p_{i,n} = \hat{F}[x(i); \eta, \theta, \kappa]$$
denotes the \(i\)'th plotting position for a sample of size \(n\); that is, a
nonparametric estimate of the value of \(F\) at \(x(i)\). Typically,
plotting positions have the form:
$$p_{i,n} = \frac{i-a}{n+b} \;\;\;\; (2)$$
where \(b > -a > -1\). In their simulation studies, Castillo and Hadi (1994)
used a=0.35, b=0.
Since \(j\) is arbitrary in the above set of equations (1), denote the solutions to these equations by: $$\hat{\eta}_j, \hat{\theta}_j, \hat{\kappa}_j$$ There are thus \(n-2\) sets of estimates.
Castillo and Hadi (1994) show that the estimate of the shape parameter, \(\kappa\), is the solution to the equation: $$\frac{x(j) - x(n)}{x(1) - x(n)} = \frac{1 - A_{jn}^\kappa}{1 - A_{1n}^\kappa} \;\;\;\; (3)$$ where $$A_{ik} = C_i / C_k \;\;\;\; (4)$$ $$C_i = -log(p_{i,n}) \;\;\;\; (5)$$ Castillo and Hadi (1994) show how to easily solve equation (3) using the method of bisection.
Once the estimate of the shape parameter is obtained, the other estimates are given
by:
$$\hat{\theta}_j = \frac{\hat{\kappa}_j [x(1) - x(n)]}{(C_n)^{\hat{\kappa}_j} - (C_1)^{\hat{\kappa}_j}} \;\;\;\; (6)$$
$$\hat{\eta}_j = x(1) - \frac{\hat{\theta}_j [1 - (C_1)^{\hat{\kappa}_j}]}{\hat{\kappa}_j} \;\;\;\; (7)$$
Second Stage
Apply a robust function to the \(n-2\) sets of estimates obtained in the
first stage. Castillo and Hadi (1994) suggest using either the median or the
least median of squares (using a column of 1's as the predictor variable;
see the help file for lmsreg in the package MASS). Using
the median, for example, the final distribution parameter estimates are
given by:
$$\hat{\eta} = Median(\hat{\eta}_2, \hat{\eta}_3, \ldots, \hat{\eta}_{n-1})$$
$$\hat{\theta} = Median(\hat{\theta}_2, \hat{\theta}_3, \ldots, \hat{\theta}_{n-1})$$
$$\hat{\kappa} = Median(\hat{\kappa}_2, \hat{\kappa}_3, \ldots, \hat{\kappa}_{n-1})$$
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