Lower bounds to the accuracy of inference on heavy tails

Abstract

The paper suggests a simple method of deriving minimax lower bounds to the accuracy of statistical inference on heavy tails. A well-known result by Hall and Welsh (Ann. Statist. 12 (1984) 1079-1084) states that if $\hat{\alpha}_n$ is an estimator of the tail index $\alpha_P$ and $\{z_n\}$ is a sequence of positive numbers such that $\sup_{P\in{\mathcal{D}}_r}\mathbb{P}(|\hat{\alpha}_n-\alpha_P|\ge z_n)\to0$, where ${\mathcal{D}}_r$ is a certain class of heavy-tailed distributions, then $z_n\gg n^{-r}$. The paper presents a non-asymptotic lower bound to the probabilities $\mathbb{P}(|\hat{\alpha}_n-\alpha_P|\ge z_n)$. We also establish non-uniform lower bounds to the accuracy of tail constant and extreme quantiles estimation. The results reveal that normalising sequences of robust estimators should depend in a specific way on the tail index and the tail constant.