Minimax multiple t-tests for comparing k normal populations with a control
Let \(\pi_ 1,...,\pi_ k\) be k normal populations with unknown means \(\theta_ 1,...,\theta_ k\), and a common unknown variance \(\sigma^ 2>0\). Based on independent samples of sizes \(n_ 1,...,n_ k\), the populations are to be partitioned into two sets, where the first one contains all \(\pi_ i\) with \(\theta_ i\geq \theta_ 0\), and where the other one contains the rest. At first it is assumed that \(\theta_ 0\) is known. Under an additive \('a_ i-b_ i'\) loss function a minimax procedure is derived which is of a simple natural form. The proof of minimaxity makes use of the Bayes approach and involves a sequence of nonsymmetric priors, which play a similar role as a least favorable prior in simpler problems. Analogous results are presented for the case that \(\theta_ 0\) is not known. In this case, a control normal population is assumed to exist from which an additional sample of size \(n_ 0\) can be drawn.
scientific article; zbMATH DE number 3915398
scientific article