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 ${\sup }_{P\in {\mathcal{D}}_r}\mathbb{P}(|{\hat{\alpha }}_n-{\alpha }_P|\ge z_n)\to 0$, 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.

arXiv commentPublished in at http://dx.doi.org/10.3150/13-BEJ512 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)