Optimal choice of sample fraction in extreme-value estimation
Let \(X_ 1, \dots, X_ n\) be an iid sample from an unknown distribution \(F\). A distribution \(F\) is in the maximum domain of attraction of an extreme value distribution \(G_ \gamma\) if there exist constants \(c_ n\), \(d_ n\) such that \[ c_ n^{-1} \bigl( \max (X_ 1, \dots, X_ n)-d_ n \bigr) @>d>>G_ \gamma. \] The distribution function \(G_ \gamma(x)\) is then of the form \(\exp \{-(1 + \gamma x)^{-1/ \gamma}\}\) with the natural choice for \(x\)-values and \(\gamma \in {\mathcal R}\). The authors study in detail the asymptotic bias of moment estimators \(\widehat \gamma_ n\) for the extreme value index \(\gamma\) under natural conditions such as regular variation of the generalised inverse of \(1/(1- F)\) and its modifications and generalizations. They also consider the trade-off between bias and variance of \((\widehat \gamma_ n - \gamma)\). In particular, they determine the fraction of the upper order statistics of the sample which minimises \(\text{var} (\widehat \gamma_ n - \gamma)\).
regular variation of the generalised inverse
scientific article; zbMATH DE number 562369