Lower bounds to the accuracy of inference on heavy tails

Article, Preprint, Other literature type English OPEN
Novak, S.Y. (2014)
  • Publisher: Bernoulli Society for Mathematical Statistics and Probability
  • Journal: (issn: 1350-7265)
  • Related identifiers: doi: 10.3150/13-BEJ512
  • Subject: Mathematics - Statistics Theory | heavy-tailed distribution | lower bounds

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 $\sup_{P\in\mathcal{D}_{r}}\mathbb{P}(|\hat{\alpha}_{n}-\alpha_{P}|\ge z_{n})\to0$, 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.
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