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

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