A general class of estimators of the extreme value index
A sequence of i.i.d. variables having common distribution \(F\) is observed. This distribution belongs to the weak domain of attraction of some extreme value distribution \(G\). That means \(\displaystyle{\mathcal L} (a_n^{-1}(\max_{1\leq i\leq n} X_i-b_n))\rightarrow G\) for some sequences \(a_n\) and \(b_n\). It is well known that \(G\) has the form \[ G_{\beta}(x)=exp(-(1+\beta x)^{1/\beta}),\quad 1+\beta x>0,\quad \beta\in {\mathbb R}. \] The paper under review studies different estimators of the so-called extreme value index \(\beta\). Naturally, the estimators must use only large observations. Hence the author considers estimators of the form \(\hat{\beta}=T(Q_n)\), where \(T\) is a functional and \(Q_n\) is the empirical tail quantile function. Almost all the usual estimators in the literature of the extreme value index are of this type. The method for studying the asymptotic behavior of such estimators is based on a theorem which deals with the Gaussian approximation of the empirical tail quantile function. The results have different proofs if \(\beta>0\) or \(\beta\leq 0\). In section 3, the case \(\beta>0\) is considered obtaining asymptotic normality under several conditions for the distribution function \(F\), and also a differentiability hypothesis for the functional \(T\). In the case \(\beta \leq 0\) the author needs to impose a condition on the local behavior of the functional \(T\) in a neighborhood of the constant function \(1\) to obtain asymptotic normality.
scientific article; zbMATH DE number 1115612