Asymptotically Most Powerful Tests in Markov Processes

Abstract

Wald [8] treated the problem of testing $H_0:\theta = \theta_0$ against one-sided alternatives by giving conditions under which tests based on the maximum likelihood estimator are asymptotically most powerful. He defines a sequence of tests $\{\lambda_n\}$ to be an asymptotically most powerful test of $H_0$ against $H_1:\theta > \theta_0$, on level of significance $\alpha$, if for any other sequence of tests $\{\omega_n\}$ of level $\alpha$, \begin{equation*} \tag{1.1} \lim \sup \lbrack \sup (\varepsilon_\theta\omega_n - \varepsilon_\theta\lambda_n; \theta > \theta_0, \theta \varepsilon \Theta) \rbrack = 0,\end{equation*} where $\Theta$ is the parameter space. A similar expression holds for testing $H_0$ against $H_2:\theta < \theta_0$. Wald's regularity conditions on the population density are quite strong. The maximum likelihood estimator, being the value which maximizes the likelihood function, is required to be a consistent estimator in the probability sense and the consistency must be uniform over certain intervals in $\Theta$. Also, his conditions imply that the centered and scaled version converges uniformly in law, over certain intervals in $\Theta$, to the standard normal. Wald formulates tests in terms of the maximum likelihood estimate. In the present work, we extend Wald's results in two directions. First, the regularity conditions are substantially weakened through use of the techniques of LeCam and the tests need not be based on the maximum likelihood estimate. Secondly, tests concerning the parameter in the joint distribution of the random variables involved are shown to be asymptotically most powerful when the observations arise from a stationary Markov process. In order that an $\alpha$-level test exist for any $\alpha$, it was necessary to consider tests which are possibly randomized. Section 2 contains the basic assumptions on the Markov process and the preliminary results appear in Section 3. The results through Section 3 hold for a $k$-dimensional parameter space and are presented in this general formulation. The main results are presented as Theorems 4.1 and 4.2 in Section 4. The following section treats the special case of independent identically distributed random variables. Four examples are presented in Section 6. In order to avoid unnecessary repetition in this paper, all limits will be taken as the sequence $\{n\}$ of positive integers, or a subsequence, converges to infinity. The present authors hope to be able to report soon on results concerning $k$-dimensional parameter versions of the main theorems in this paper.