Asymptotic considerations for selecting the best component of a multivariate normal population (Q1087262)

The problem discussed is that of selecting the component with the largest mean value in a k-variate normal population with unknown variance- covariance matrix with positive correlations. Let \(\mu_{[1]}\leq \mu_{[2]}\leq...\leq \mu_{[k]}\) be the ordered means of a k-variate normal population and let \(\Omega (\delta^*)=\{(\mu,\Sigma):\Sigma\) positive definite and \(\mu_{[k]}-\mu_{[k-1]}\geq \delta^*\}.\) The goal is to select the component with mean equal to \(\mu_{[k]}\) in such a way that the probability of correct selection is at least equal to \(P^*\) whenever (\(\mu,\Sigma)\in \Omega(\delta^*)\). Two-stage and sequential procedures having the \(P^*\) property are considered. The rates of convergence of the probability of correct selection of the procedures as \(\delta^*\to 0\) are presented. For a proposed sequential procedure, some results of the average sample number are presented.