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Statistica Sinica 25 (2015), 1133-1144

ADAPTIVE AND MINIMAX OPTIMAL ESTIMATION OF
THE TAIL COEFFICIENT
Alexandra Carpentier and Arlene K. H. Kim
University of Cambridge

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 n)β∕(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.

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