Lower bounds to the accuracy of inference on heavy tails

Novak, Serguei ORCID: https://orcid.org/0000-0001-7929-7641 (2014) Lower bounds to the accuracy of inference on heavy tails. Bernoulli, 20 (2) . pp. 979-989. ISSN 1350-7265 [Article] (doi:10.3150/13-BEJ512)

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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 α^n is an estimator of the tail index αP and {zn} is a sequence of positive numbers such that supP∈DrP(|α^n−αP|≥zn)→0, where Dr is a certain class of heavy-tailed distributions, then zn≫n−r. The paper presents a non-asymptotic lower bound to the probabilities P(|α^n−αP|≥zn). 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.

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