Cramér type large deviations for Studentized U-statistics
type: Item
instance of: scholarly article
title: Cramér type large deviations for Studentized U-statistics
zbMATH Open document ID: 0617.62015
DOI: 10.1007/BF01897811
author: M. Vandemaele
author: Noël Veraverbeke
published in: Metrika
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/176001
review text: Let \(U_ n=\left( \begin{matrix} n\\ 2\end{matrix} \right)^{-1}\sum \sum_{1\leq i<j\leq n}h(X_ i,X_ j)\) denote a U-statistic with kernel h of degree 2, based on n independent and identically distributed random variables \(X_ 1,X_ 2,...,X_ n\). Assume that E h(X\({}_ 1,X_ 2)=\theta\) and let the variance of \(E[h(X_ 1,X_ 2)| X_ 1]\) be positive. If, in addition, \(E| h(X_ 1,X_ 2)-\theta |^ p\leq K^ pp^{\gamma p}\) for \(p=1,2,..\). and some constants K and \(\gamma\geq 0\) not depending on p, then \[ P(\sqrt{n}(U_ n-\theta)/S_ n>x)=[1-\Phi (x)](1+o(1)) \] uniformly in the range \(0\leq x\leq o(n^{\alpha})\), with \(\alpha =2^{-1}(3+2\gamma)^{-1}\) for \(\gamma\leq 2\) and \(\alpha =2^{-1}(4\gamma -1)^{-1}\) for \(\gamma\geq 2\). The quantity \(n^{- 1}S^ 2_ n\) is the jackknife estimator of the variance of \(U_ n\) and \(\Phi\) denotes the standard normal distribution function. For the important special case of the Student t-statistic (take \(h(x,y)=2^{- 1}(x+y))\) the authors obtain a somewhat better range for \(\alpha\).
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4001182
zbMATH Keywords: Cramér type large devations
zbMATH Keywords: U-statistic
zbMATH Keywords: jackknife estimator
zbMATH Keywords: standard normal distribution
zbMATH Keywords: Student t-statistic
MaRDI profile type: Publication
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OpenAlex ID: W2091837782
description: scientific article; zbMATH DE number 4001182