Lower bounds to the accuracy of inference on heavy tails
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چکیده
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 α̂n is an estimator of the tail index αP and {zn} is a sequence of positive numbers such that sup P∈Dr P(|α̂n − αP | ≥ zn) → 0, where Dr is a certain class of heavy-tailed distributions, then zn ≫ n . The paper presents a non-asymptotic lower bound to the probabilities P(|α̂n−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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