Asymptotic consistency for subset selection procedures satisfying the \(P^ *\)-condition
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.
scientific article; zbMATH DE number 3988460
scientific article