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.