On the Power of One-tailed Tests when the Test Statistic is Monotone in the Hazard Rate Stochastic Order

Accepted June 2026

Abstract

A direct consequence of the Karlin-Rubin Theorem is that, under appropriate assumptions on the family of distributions corresponding to the statistic used, one-tailed tests for a population parameter based on one-sided rejection regions are at least as powerful as those based on interval rejection regions. Fundamentally, it requires the family of distributions associated with the statistic to be increasing with the parameter in the likelihood ratio stochastic order. In this note it is shown that the same conclusion can be obtained by replacing the likelihood ratio order with the weaker hazard rate order, thus extending its validity to more cases.