Adaptive and minimax optimal estimation of the tail coefficient

We consider the problem of estimating the tail index \alpha of a distribution satisfying a (\alpha, \beta) second-order Pareto-type condition, where \beta is the second-order coefficient. When \beta is available, it was previously proved that \alpha can be estimated with the oracle rate n^{-\beta/(2\beta+1)}. On the contrary, when \beta is not available, estimating \alpha with the oracle rate is challenging; so additional assumptions that imply the estimability of \beta are usually made. In this paper, we propose an adaptive estimator of \alpha, and show that this estimator attains the rate (n/\log\log n)^{-\beta/(2\beta+1)} without a priori knowledge of \beta and any additional assumptions. Moreover, we prove that this (\log\log n)^{\beta/(2\beta+1)} factor is unavoidable by obtaining the companion lower bound.