On Optimal Decision Rules for Signs of Parameters

Abstract

The problem of deciding the signs of $k$ parameters $(\theta_1, \cdots, \theta_k) \equiv \mathbf{theta}$ based on $(\hat{\theta}_1, \cdots, \hat{\theta}_k) \sim N(\mathbf{\theta,\Sigma})$ such that $p_\mathbf{\theta} \{$\text{any error$\} \leq \alpha \forall \mathbf{\theta}$ is discussed by Bohrer and Schervish (1980). They characterize a desirable class of procedures called locally optimal. For the case $k = 2, \mathbf{\Sigma = I}$, and $\alpha \leq \frac{1}{3}$, they present a particular rule from this class called the double cross. In this paper, we address the problem of selecting a best rule from among all locally optimal rules when $k = 2$ and $\mathbf{\Sigma = I}$. When $\alpha \leq \frac{1}{3}$, the double cross is shown to be an attractive choice. Other rules are obtained for higher values of $\alpha$. We also examine a more general optimization criterion than the one used by Bohrer and Schervish and obtain different optimal rules for several classes of problems. The optimal rule corresponding to one of these classes has no two-decision region. A modification of the formulation is offered under which a well-known rule (with two decision regions) emerges as the unique optimal procedure.