On the rate of convergence in the central limit theorem for signed rank statistics
type: Item
instance of: scholarly article
title: On the rate of convergence in the central limit theorem for signed rank statistics
zbMATH Open document ID: 0568.62018
DOI: 10.1016/0196-8858(85)90003-X
author: Stefan S. Ralescu
author: Madan L. Puri
published in: Advances in Applied Mathematics
publication date: 1985-01-01T00:00:00Z
review text: This paper deals with the rate of convergence of the distribution functions of signed rank statistics for one sample to the normal d.f.: \[ S^+_ N=N^{-1}\sum^{N}_{i=1}1_{[0,\infty [}(X_ i)\cdot \Psi (R_ i/N+1) \] where \(R_ i\) is the rank of \(| X_ i|\) among \(| X_ 1|,...,| X_ N|.\) It generalizes known results to non-standard cases: when the underlying observations are independent but not identically distributed, and for score generating functions \(\Psi\) which are unbounded and not necessarily monotonous (including the normal scores statistic). The approach is the same as the Chernoff-Savage, expressing the statistic \(S^+_ N\) as an integral of the empirical distribution functions; so, they state a generalization of Dvoretzky-Kiefer-Wolfovitz' theorem to the non identically distributed case. As a corollary of the proof, a strong law for \(S^+_ N\) and a functional central limit theorem are given.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 3907555
zbMATH Keywords: asymptotic normality
zbMATH Keywords: Chernoff-Savage approach
zbMATH Keywords: cumulative distribution function
zbMATH Keywords: one-sample rank order statistics
zbMATH Keywords: rate of convergence
zbMATH Keywords: distribution functions of signed rank statistics
zbMATH Keywords: integral of the empirical distribution functions
zbMATH Keywords: generalization of Dvoretzky-Kiefer- Wolfovitz' theorem
zbMATH Keywords: non identically distributed case
zbMATH Keywords: functional central limit theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3907555