Spacings-ratio empirical processes
type: Item
instance of: scholarly article
title: Spacings-ratio empirical processes
zbMATH Open document ID: 1240.60036
author: P. Deheuvels
author: Gérard Derzko
published in: Periodica Mathematica Hungarica
publication date: 2011-12-19T00:00:00Z
review text: The paper under review is to consider an empirical process based on ratios of selected pairs of spacings generated by independent samples of arbitrary sizes, and to show that this empirical process converges to a mean centered Brownian bridge provided both samples are uniformly distributed on intervals of equal lengths. Let \(X=\{X_1, \cdots, X_{n_1}\}\) and \(Y=\{Y_1, \cdots, Y_{n_2}\}\) be two independent samples of sizes \(n_1\) and \(n_2\), composed of i.i.d.\ random variables generated by the continuous distribution function \(F_X\) and \(F_Y\). A motivation of this paper under review is to construct nonparametric tests of the null hypothesis (H.0) against the alternative (H.1). \noindent (H.0): There exists a \(\mu \in {\mathbb R}\) such that \(F_X(t) = F_Y(t-\mu)\) for all \(t \in {\mathbb R}\). \noindent (H.1): There does not exist any \(\mu \in {\mathbb R}\) such that \(F_X(t) = F_Y(t-\mu)\) for all \(t \in {\mathbb R}\). The Wilcoxon test, the Mann-Whitney test and the Hodges-Lehmann shift estimator are used to estimate the shift value \(\mu\) when (H.0) holds or to test the assumption that \(\mu =0\) against the alternative that \(\mu\in {\mathbb R}\) is arbitrary. There is no general distribution-free nonparametric methodology to test (H.0) against a general alternative. A main result Theorem 1.1 is that under (H.0) and the distribution of the sample components are uniform on intervals of \({\mathbb R}\), \(\{\gamma_n (u): 0\leq u \leq 1\}\) can be approximated by mean centered Brownian bridges \(\{B(u)-12u(1-u)\int_0^1B(s)ds: 0\leq u \leq 1\}\), where \(\{B(u): 0\leq u \leq 1\}\) is a Brownian bridge. The limiting process of \(\gamma_n (\cdot)\) under (H.0) depends upon \(F(\cdot)\). The main arguments of Theorem 1.1 are presented in Section 3, according to the study of the empirical processes based on sums and ratios of independent exponentially distributed random variables. In Section 4 the authors give the construction of the Gaussian processes for a…
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 62G10
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 5990588
zbMATH Keywords: empirical processes
zbMATH Keywords: Gaussian processes
zbMATH Keywords: weak invariance principles
zbMATH Keywords: limit laws
zbMATH Keywords: spacings
zbMATH Keywords: order statistics
zbMATH Keywords: weak laws
zbMATH Keywords: two sample tests
zbMATH Keywords: nonparametric tests
MaRDI profile type: Publication
full work available at URL: s10998-010-3121-0
OpenAlex ID: W2029476466
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DOI: 10.1007/S10998-010-3121-0
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description: scientific article; zbMATH DE number 5990588