On nonparametric profile analysis of several multivariate samples
type: Item
instance of: scholarly article
title: On nonparametric profile analysis of several multivariate samples
zbMATH Open document ID: 0564.62031
DOI: 10.1016/0378-3758(84)90004-1
author: Vasant P. Bhapkar
published in: Journal of Statistical Planning and Inference
publication date: 1984-01-01T00:00:00Z
review text: Let \((X_{it}^{(1)},...,X_{it}^{(p)})\), \(t=1,2,...,n_ i\) be a random sample from the ith population with p-variate continuous distribution function \(F_ i\) for \(i=1,2,...,k\). Let \(\{X_{it}^{(\alpha)}\), \(t=1,2,...,n_ i\), \(i=1,2,...,k\}\) be arranged in increasing order and denote by \(R_{it}^{(\alpha)}\) the rank of \(X_{it}^{(\alpha)}\) in this increasing sequence. Put \(F=(F_ 1,...,F_ k)\) and define \(\xi_{iu}^{(\alpha)}(F)\) by the probability that the \(u^{th}\) smallest in the ranking of \(\{X_{it}^{(\alpha)}\), \(t=1,2,...,n_ i\), \(i=1,2,...,k\}\) is from the \(i^{th}\) sample. The population profiles are said to be parallel with respect to the rank order of \(N=\sum^{k}_{i=1}n_ i\) observations if \(\xi_{iu}^{(1)}(F)=\xi_{iu}^{(2)}(F)=...=\xi_{iu}^{(p)}(F)\) for all \(u=1,2,...,N\) and \(i=1,2,...,k\). For a given system of scores \(\{s_ u\), \(u=1,2,...,N\}\), put \(\eta_ i^{(\alpha)}(F)=\sum^{N}_{u=1}s_ u\xi_{iu}^{(\alpha)}(F)/n_ i\). Then the \(i^{th}\) population profile is said to be parallel to the axis if \(\eta_ i^{(1)}(F)=\eta_ i^{(2)}(F)=...=\eta_ i^{(p)}(F).\) Asymptotically distribution-free tests for parallelism of profile for a subset of p variables and a subset of k populations are discussed, including linear rank statistics and generalized U-statistics. Their asymptotic powers and consistency of a class of tests are developed.
Mathematics Subject Classification ID: 62G10
Mathematics Subject Classification ID: 62H15
zbMATH DE Number: 3899954
zbMATH Keywords: multivariate samples
zbMATH Keywords: location profiles
zbMATH Keywords: scalar profiles
zbMATH Keywords: profile analysis
zbMATH Keywords: population profiles
zbMATH Keywords: Asymptotically distribution-free tests
zbMATH Keywords: parallelism of profile
zbMATH Keywords: linear rank statistics
zbMATH Keywords: generalized U-statistics
zbMATH Keywords: asymptotic powers
zbMATH Keywords: consistency
reviewed by: Nariaki Sugiura
MaRDI profile type: Publication
full work available at URL: 0378-3758(84)90004-1
OpenAlex ID: W2061290829
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description: scientific article; zbMATH DE number 3899954