Statistica Sinica 25 (2015), 1133-1144
Abstract: We consider the problem of estimating the tail index α of a distribution satisfying a (α,β) second-order Pareto-type condition, where β is the second-order coefficient. When β is available, it was previously proved that α can be estimated with the optimal rate n-β∕(2β+1). When β is not available, estimating α with the optimal rate is challenging ; so additional assumptions that imply the estimability of β are usually made. We propose an adaptive estimator of α, and show that this estimator attains the rate n∕log log n-β∕(2β+1) without a priori knowledge of β or additional assumptions. Moreover, we prove that a log log nβ∕(2β+1) factor is unavoidable by obtaining the companion lower bound.
Key words and phrases: Adaptive estimation, extreme value index, minimax optimal bounds, Pareto-type distributions.