Asymptotic relative efficiency of tests at the boundary of regular statistical models

Abstract

Consider a sequence of tests which are asymptotically equivalent to some score tests for a regular model. In the present paper we address the question what's their power and asymptotic relative efficiency (ARE) at the boundary of the statistical model when the Fisher information becomes infinite but still local asymptotic normality (LAN) holds. The boundary models are frequently called almost regular. It is shown that almost regular models and efficient procedures for the almost regular case are completely different from the regular situation. The ARE of regular tests collapses at the boundary. A specific example is the two-sample life time regression model given by Weibull distributions. Here, the Wilcoxon two-sample and Savage rank tests (log-rank test) have ARE=0 at the boundary. The results are based on a uniform LAN-expansion for master models specified by tangents. This part is of own interest.