On the Unconditional Optimality of Conditional Tests

Abstract

When an ancillary statistic is present, assuming that the loss resulting from a wrong decision depends on the ancillary, it is shown that the best conditional size­α test for a simple H_o against a simple H₁ minimizes the unconditional risk under H₁ subject to an upper bound on the risk under H_o, provided the loss depends on the ancillary in a certain special manner. Considering the case when the conditional distribution has monotone likelihood ratio, it is next shown that this special type of Joss function can be so chosen as to possess two intuitively reasonable properties.