The Bootstrap Methodology in Statistics of Extremes—Choice of the Optimal Sample Fraction

The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the tail index γ, usually performed on the basis of the largest k order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive estimation of γ. We shall be here mainly interested in the use of the bootstrap methodology to estimate γ adaptively, and although the methods provided may be applied, with adequate modifications, to the general domain of attraction of Gγ, γ ∈ ℝ, we shall here illustrate the methods for heavy right tails, i.e. for γ > 0. Special relevance will be given to the use of an auxiliary statistic that is merely the difference of two estimators with the same functional form as the estimator under study, computed at two different levels. We shall also compare, through Monte Carlo simulation, these bootstrap methodologies with other data-driven choices of the optimal sample fraction available in the literature.