On optimal decision rules for signs of parameters (Q1089708)

Let X have a bivariate normal distribution with known covariance matrix and expectation \(\theta\). Interest is on an optimal decision rule for the signs of the components of \(\theta\) which is based on X. Optimality means to maximize the expected number of correct decisions - when \(\theta\) tends to zero - under all decision rules which are symmetric, upper convex and satisfying \(P_{\theta}(no\) incorrect decision) \(\geq 1- \alpha\) for all \(\theta\). The problem can be formulated as a linear programming problem. Its solutions are discussed using the Neyman-Pearson lemma. The results extend earlier results by \textit{R. Bohrer} and \textit{M. J. Schervish}, Proc. Natl. Acad. Sci. USA 77, 52-56 (1980; Zbl 0428.62008).