On the Asymptotic Efficiency of Conditional Tests for Exponential Families

Abstract

Let $P_\eta, \eta \in \Theta \times \Gamma \subset \mathbb{R} \times \mathbb{R}^k$, be an exponential family. It is shown that the sequence of tests $(\varphi^\ast_n)_{n\in\mathbb{N}}$, where $\varphi^\ast_n, n \in \mathbb{N}$, is u.m.p. in the class of all tests similar with respect to the nuisance-parameter $\gamma$ for the hypothesis $\{P^n_{(\theta, \gamma)}: \gamma \in \Gamma\}$ against alternatives $P^n_{(\theta_1, \eta_1)}, \theta_1 > \theta, \eta_1 \in \Gamma$, is asymptotically efficient in the class $\Phi^\ast_\alpha$ of test-sequences which are asymptotically of level $\alpha$ (continuously in the nuisance-parameter). Here, asymptotic efficiency of $(\varphi^\ast_n)_{n\in\mathbb{N}}$ means that for all $\gamma \in \Gamma, t > 0$, the power of $\varphi^\ast_n$ evaluated at local alternatives $P^n_{(\theta+tn^{-1/2},\gamma)}$ asymptotically attains the upper bound given for test-sequences in $\Phi^\ast_\alpha$.