Best Tests for Testing Hypotheses About a Random Parameter with Unknown Distribution

Abstract

Let $X$ be a random variable with a family of possible distributions for $X$ indexed by $\lambda\in\Omega. \lambda$ is the realization of a random variable $\Lambda$ taking values in the space $\Omega$. For each $\lambda$, let $f_\lambda$ denote the conditional density of $X$ given $\Lambda = \lambda$ with respect to some $\sigma$-finite measure $\mu$. Let $\mathscr{G}$ be a family of possible a priori distributions $G$ for $\Lambda$. After observing $X$, we wish to test $H: \lambda\in\omega$ against $K: \lambda\in\omega'$ where $\omega$ is a subset of $\Omega$ and $\omega'$ its complement. To determine good tests for this problem, we use an analysis similar to the one of the Neyman-Pearson theory of hypothesis testing. Analogous to the type I and type II errors of the Neyman-Pearson theory are: type (i) error: $\Lambda\in\omega'$ decided and $\Lambda\in\omega$ occurs, type (ii) error: $\Lambda\in\omega$ decided and $\Lambda\in\omega'$ occurs. Analogous to the problem of finding uniformly most powerful level $\alpha$ tests is the problem: subject to: $P_G$(type (i) error) $\leqq \alpha$ for all $G\in\mathscr{G}$ minimize $P_G$(type (ii) error) uniformly for $G\in\mathscr{G}$. A test which achieves this is called a uniformly most powerful (UMP) level $\alpha$ test relative to $\mathscr{G}$. The existence of such UMP level $\alpha$ tests is proved for this hypotheses testing problem for various choices of the family of a priori distributions $\mathscr{G}$. As might be expected these results are closely related to the Neyman-Pearson theory of hypotheses testing. The second section gives four simple situations where the problem of finding UMP level $\alpha$ tests relative to a family of a priori distributions $\mathscr{G}$ reduces to an ordinary testing problem. In the third section, Theorem 1 gives for this testing problem an analogue of the concept of a least favorable distribution from the classical theory of hypotheses testing. Theorem 1 is used to prove Theorem 2 which gives the existence of a UMP level $\alpha$ test when $X$ is real-valued, $\Omega$ is a subset of the real numbers, the family of distributions indexed by $\lambda\in\Omega$ has a monotone likelihood ratio in $x$, and the family $\mathscr{G}$ satisfies a certain condition. The two theorems are applied to several examples. In the following, as always, a test (randomized) is a function $\delta$ defined on the range of $X$ which takes on values in the interval $\lbrack 0, 1\rbrack$. If $X = x$ is observed, $K$ is decided to be true with probability $\delta(x)$ and $H$ with probability $1 -\delta(x)$. For any test $\delta$ and $G\in\mathscr{G}$ we have \begin{equation*}\tag{1} P_G(\text{type (i) error of} \delta) = \int \int_\omega \delta(x)f_ \lambda(x) dG(\lambda) d \mu \quad\text{and}\end{equation*} \begin{equation*}\tag{2} P_G(\text{type (ii) error of} \delta) = \int \int_{\omega'} (1 - \delta(x))f_\lambda(x) dG(\lambda) d\mu\end{equation*} where the integral involving $X$ is over the entire space of $X$. It will often be convenient to think of $\lambda$ as a fixed but unknown parameter and the test $\delta$ as a test for the classical testing problem $H: \lambda\in\omega$ against $K: \lambda\in\omega'$. Changing the order of integration in (1) by Fubini's theorem, we have for the test $\delta$ the following relationship between the type I error of $\delta$, considered as a test for the classical problem, and the type (i) error of $\delta$, considered as a test for the problem of this paper: \begin{equation*}\tag{3} P_G(\text{type (i) error of} \delta) = \int_\omega P_\lambda (\text{type I error of} \delta) dG(\lambda).\end{equation*} In the same way, we have \begin{equation*}\tag{4} P_G(\text{type (ii) error of} \delta) = \int_{\omega'} P_\lambda (\text{type II error of} \delta) dG(\lambda).\end{equation*} We will now prove the existence of UMP level $\alpha$ tests for various families of a priori distributions.