Dekkers, Arnold L. M.; de Haan, Laurens Optimal choice of sample fraction in extreme-value estimation. (English) Zbl 0797.62016 J. Multivariate Anal. 47, No. 2, 173-195 (1993). 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 \(\text{var} (\widehat \gamma_ n - \gamma)\). Reviewer: T.Mikosch (Zürich) Cited in 67 Documents MSC: 62F12 Asymptotic properties of parametric estimators 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions 60G70 Extreme value theory; extremal stochastic processes Keywords:Hill estimator; iid sample; maximum domain of attraction; extreme value distribution; asymptotic bias of moment estimators; extreme value index; regular variation of the generalised inverse; upper order statistics PDFBibTeX XMLCite \textit{A. L. M. Dekkers} and \textit{L. de Haan}, J. Multivariate Anal. 47, No. 2, 173--195 (1993; Zbl 0797.62016) Full Text: DOI Link