Parametric Optimal Testing: Theory and Applications

Abstract

This paper puts forward a uniÖed framework for asymptotically optimal tests in parametric models and applies the new theory to two tests. The Örst test is a classical test in locally asymptotically normal (LAN) models, but the assumptions are weakened and a lacuna in the literature is Ölled. The second test concerns the location of the threshold point. The main result here is that the optimal test in the weighted average power sense is based on the posterior odds which depends on the prior on the local parameter space and is not unique. Furthermore, the likelihood ratio test is not asymptotically equivalent to the posterior odds and there is a discrete component in its asymptotic distribution. Since the asymptotic distribution of the posterior odds is not pivotal to the true value under the null hypothesis, a parametric bootstrap is used to Önd asymptotic critical values. The results in the second test are very di§erent from those in classical LAN models.