An optimal multiple decision rule for signs of parameters

The problem of simultaneously deciding the signs ofp parameters based on normally distributed estimators is considered. A decision that a parameter is too close to zero to decide its sign is allowed in order that the probability of at least one incorrect decision can be kept less than a preassigned value. Concepts of upper-convexity and local optimality are defined, and forp = 2 or 3 the locally optimal rules in the class of upper-convex rules are found. Critical values and operating characteristics are given for the locally optimal rule and some other plausible rules whenp = 2.