Bahadur-Kiefer-type processes
type: Item
instance of: scholarly article
title: Bahadur-Kiefer-type processes
zbMATH Open document ID: 0712.60028
author: P. Deheuvels
author: David M. Mason
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: Let \(\alpha_ n(s)=n^{1/2}(F_ n(s)-s)\) (0\(\leq s\leq 1\), \(n=1,2,...)\) resp. \(\beta_ n(s)\) be the uniform empirical resp. quantile process. Further let \(R_ n(s)=\alpha_ n(s)+\beta_ n(s)\) and \(b_ n=n^{1/4}(\log n)^{-1/2}\). \textit{J. Kiefer} [Deviations between the sample quantile process and the sample d.f., in: Proc. Symp., Indiana Univ., Bloomington, Ind. (1969), 299-319 (1970)] claimed that \[ b_ n\sup | R_ n(s)| /(\sup | \beta_ n(s)|)^{1/2}\to 1\quad a.s. \] The authors give a proof of this statement. They present an analogous result replacing \(\alpha_ n\) resp. \(\beta_ n\) by a standardized partial sum resp. renewal process. Weighted versions of the above results are also presented.
reviewed by: Pál Révész
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 4172049
zbMATH Keywords: Bahadur representation
zbMATH Keywords: empirical processes
zbMATH Keywords: quantile process
zbMATH Keywords: sample quantile
zbMATH Keywords: renewal process
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990852
OpenAlex ID: W2073358569
DOI: 10.1214/AOP/1176990852
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description: scientific article; zbMATH DE number 4172049