A location invariant Hill-type estimator
A new estimator, under a semiparametric approach, which behaves similarly to Hill's estimator, see \textit{B.M. Hill}, Ann. Stat. 3, 1163--1174 (1975; Zbl 0323.62033), but which is invariant to location transformations, is studied. Let \(X_1,X_2,\ldots,X_ n\) be independent random variables with common distribution function \(F\) and let \(X_{(1,n)},X_{(2,n)},\ldots,X_{(n,n)}\) be the associated increasing order statistics. Suppose that \(F\) is in the domain of attraction of an extreme value distribution \(G_{\gamma}\) with \[ G_{\gamma}(z)=\begin{cases} \exp(-(1+\gamma z)^{-1/\gamma}),& \text{for } 1+\gamma z>0, \text{ if } \gamma\neq0\\ \exp(-\exp(-z)),& \text{for } z\in R, \text{ if } \gamma=0.\end{cases} \] The author proposes the following estimator. For \(\gamma>0\) and with an integer \(k_0\) (smaller than \(k\)) define \[ \widehat\gamma^{H}_{n}(k_0,k):= k_0^{-1} \sum_{i=0}^{k_0-1} \log (X_{(n-i,n)}-X_{(n-k,n)})/ (X_{(n-k_0,n)}-X_{(n-k,n)}). \] If \(k=k(n)\to\infty\) and \(k_0=k_0(n)\to\infty\), as \(n\to\infty\), such that \(k/n\to0\) and \(k_0/k\to0\), then \(\lim_{n\to\infty} \widehat\gamma^{H}_{n}(k_0,k)=\gamma\) in probability. It is proved that under some conditions \(\sqrt{k_0} \{\widehat\gamma^{H}_{n} (k_0,k)-\gamma\}@> d>> Z\), as \(n\to\infty\), where \(Z\) is normally distributed with mean zero and variance \(\gamma^2\). This new Hill-type estimator is location invariant. A preliminary study is carried out, with the main objective of comparing this new procedure with some other traditional estimators, namely moment and Pickands', with the consideration of locations different from zero.
scientific article; zbMATH DE number 2118660