Optimal Choice of Sample Fraction in Extreme-Value Estimation

We study the asymptotic bias of the moment estimator [gamma]n for the extreme-value index [gamma] [set membership, variant] 5 under quite natural and general conditions on the underlying distribution function. Furthermore the optimal choice for the sample franction in estimating [gamma] is considered by minimizing the mean squared error of [gamma]n - [gamma]. The results cover all three limiting types of extreme-value theory. The connection between statistics and regular variation and [Pi]-variation is handled in a systematic way.