Effect on the Minimal Complete Class of Tests of Changes in the Testing Problem

Abstract

A question of interest in connection with many statistical problems is the following: Does a slight change in the problem result in a different answer? Here the effect of changes in the testing problem on the minimal complete class of tests is investigated. The effects of such changes are found to be different for the two families of distributions considered: The discrete multivariate exponential family and the continuous multivariate exponential family. In Section 2, it is shown that with respect to the discrete exponential family, the minimal complete class of tests for a standard testing problem is minimal complete for a wide variety of related problems. In Section 3, an example is given showing that with respect to the continuous exponential family, on the other hand, the minimal complete class of tests for a standard problem is not necessarily minimal complete for a slight variation of this problem. Tests that are admissible for the standard problem are not necessarily admissible for the variation. Partly in a general decision theoretic framework and partly with respect to specific examples, Hoeffding [2] has discussed the effect of changes in the family of probability distributions on the minimax solution and other optimal solutions. He has also given key references to the extensive literature on the performance of standard procedures for families of probability distributions not satisfying all the assumptions under which the standard procedures were derived. Workers in this area have primarily concentrated on the effect of changes in the probability model on a single solution rather than on a class of solutions, for example, the class of admissible solutions, as we do here. We recall some basic ideas. Consider the probability structure $(\mathcal{X, A,} P, \Omega)$ where $\mathcal{X}$ and $\Omega$ are sets, $\mathcal{A}$ is a $\sigma$-field of subsets of $\mathcal{X}$, and for each $\theta$ in $\Omega$, $P_{\theta}$ is a probability measure on $\mathcal{A}$. Relative to the above structure, a testing problem is an ordered pair $(\omega_0, \omega_1)$ of disjoint subsets of $\Omega$. A test $\varphi$ is a function from $\mathcal{X}$ into $\lbrack 0, 1\rbrack$ measurable with respect to $\mathcal{A}$. The test $\varphi$ is used in the following way: A random element $X$ with values in $\mathcal{X}$ having $P_{\theta}$ as its probability distribution is observed. If $x$ is the outcome then the hypothesis $H: \theta \varepsilon \omega_0$ is rejected with probability $\varphi(x)$ in favor of the alternative $A: \theta \varepsilon \omega_1$. If the test $\varphi$ is used and $\theta$ is the parameter, then the probability that $H$ is rejected is $E_{\theta\varphi} = \int_{\mathcal{X}} \varphi(x) dP_{\theta}(x)$. If $\varphi$ and $\varphi^{\ast}$ are tests, then $\varphi$ is at least as good as $\varphi^{\ast}$ if $E_{\theta\varphi} - E_{\theta\varphi^\ast} \leqq 0$ for $\theta$ in $\omega_0, \geqq 0$ for $\theta$ in $\omega_1$. The test $\varphi$ is better than $\varphi^{\ast}$ if $\varphi$ is at least as good as $\varphi^{\ast}$ but $\varphi^{\ast}$ is not at least as good as $\varphi$. A test $\varphi$ is admissible if there is no test better than $\varphi$. A class of tests is complete if to each test not in the class there corresponds a test in the class which is better. A class is minimal complete if it is complete and no proper subclass is complete. The notions, "essentially complete" and "minimal essentially complete," are defined similarly with "at least as good" substituted for "better."