Asymptotic consistency for subset selection procedures satisfying the \(P^ *\)-condition
type: Item
instance of: scholarly article
title: Asymptotic consistency for subset selection procedures satisfying the \(P^ *\)-condition
zbMATH Open document ID: 0611.62022
DOI: 10.1016/0378-3758(86)90142-4
author: Jan F. Bjørnstad
published in: Journal of Statistical Planning and Inference
publication date: 1986-01-01T00:00:00Z
review text: Let \(\pi_ 1,...,\pi_ k\) be k populations characterized by \(\theta_ 1,...,\theta_ k\) respectively with possible nuisance parameter \(\sigma\). The ordered \(\theta_ i\) are denoted by \(\theta_{(1)}\leq...\leq \theta_{(k)}\). The problem is to select a random nonempty subset from the k populations such that the probability that the selected subset includes the best population \(\pi_{(k)}\) (or \(\pi_{(1)})\), the population with \(\theta =\theta_{(k)}\) (or \(\theta =\theta_{(1)})\) is at least equal to \(P^*\). A subset which includes the best population is called a correct selection. Assume that \(\theta_ i\) is estimated by \(X^ n_ i\) and \(\sigma\) by \(S^ n\). Let \(X^ n=(X^ n_ 1,...,X^ n_ k)\). A subset selection procedure is given by \[ \delta_ n(a| x,s)=P\{selecting\quad \pi_ i,\quad i\in a| \quad X^ n=x,\quad S^ n=s\}, \] and the decision space is \({\mathcal A}=\{a:\) \(a\subseteq \{1,...,k\}\}\). \(\sum_{a}\delta (a| x,s)=1\). For classes of loss functions, asymptotic consistency of a decision procedure is defined as the property that the risk of the procedure tends to minimum loss as the sample size tends to infinity. Necessary and sufficient conditions are given for pointwise and uniform consistency. The loss functions considered are of the form \[ \ell (\theta,a)=\alpha (| a|)\sum_{i\in a}\ell^ 0(\theta_{(k)}- \theta_ i), \] \[ \ell (\theta,a)=\alpha (| a|)\sum_{i\in a}\ell^ 0(\theta_{(k)}/\theta_ i-1). \] Here \(| a|\) denotes the size of the selected subset a, and \(\ell^ 0\) is continuous, nondecreasing with \(\ell^ 0(0)=0\) and \(\ell^ 0(x)>0\) if \(x>0.\) The paper considers the consistency of various known selection procedures of normal and binomial populations, selection procedures of multinomial cells and procedures for selecting large Mahalanobis distances of multivariate normal populations.
Mathematics Subject Classification ID: 62F07
Mathematics Subject Classification ID: 62C99
zbMATH DE Number: 3988460
zbMATH Keywords: Gupta's procedure
zbMATH Keywords: minimax procedures
zbMATH Keywords: large sample theory
zbMATH Keywords: selecting the best population
zbMATH Keywords: subset selection procedure
zbMATH Keywords: loss functions
zbMATH Keywords: asymptotic consistency
zbMATH Keywords: risk
zbMATH Keywords: minimum loss
zbMATH Keywords: Necessary and sufficient conditions
zbMATH Keywords: pointwise and uniform consistency
zbMATH Keywords: normal
zbMATH Keywords: binomial populations
zbMATH Keywords: multinomial cells
zbMATH Keywords: Mahalanobis distances
zbMATH Keywords: multivariate normal populations
reviewed by: Umesh D. Naik
MaRDI profile type: Publication
full work available at URL: 0378-3758(86)90142-4
OpenAlex ID: W1487347168
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description: scientific article; zbMATH DE number 3988460