910 research outputs found
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
On limiting cluster size distributions for processes of exceedances for stationary sequences
It is well known that, under broad assumptions, the time-scaled point process
of exceedances of a high level by a stationary sequence converges to a compound
Poisson process as the level grows. The purpose of this note is to demonstrate
that, for any given distribution G on the natural numbers, there exists a
stationary sequence for which the compounding law of this limiting process of
exceedances will coincide with G.Comment: 6 pages, no figure
Poisson approximation
This is a survey article on the topic of Poisson approximation
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