Lower bounds to the accuracy of inference on heavy tails

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.

© 2014 ISI/BS
This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2014, Vol. 20, No. 2, 979–989. This reprint differs from the original in pagination and typographic detail.