Most Powerful Tests for Some Non-Exponential Families

Abstract

We shall be concerned with the parametric problem of testing hypotheses concerning the value of one parameter when the values of other parameters (nuisance parameters) are not specified. Neyman [6] derived under certain conditions a locally most powerful two-sided test for this problem, i.e., he gave the form of the test maximizing the second derivative of the power function with respect to the parameter of interest at the point specified by the hypothesis. Generalizations of Neyman's results were given by Scheffe [7] and Lehmann [2], using the same technique as Neyman. They were also able to prove that the tests were UMP unbiased. A new technique for dealing with these problems was introduced by Sverdrup [9] and Lehmann and Scheffe [4] where the completeness of the sufficient statistics in an exponential family of densities is used to derive UMP unbiased tests. It is stated by Lehmann and Scheffe [4] that the conditions imposed earlier imply an exponential family of densities. When no UMP unbiased test exists we have little general theory. The problem is both one of principle and of technique. Most stringest tests exist under general conditions but are difficult to derive in particular cases. Lehmann [3] proposed maximin tests. Spjotvoll [8] has given an example of the form of a maximin test when no UMP unbiased and invariant test exists. This paper is an attempt to establish some results for testing hypotheses when the probability density of the observations does not constitute an exponential family under both the hypothesis and the alternative. The assumptions made in Section 2 are satisfied if we have an exponential family under the hypothesis, but do not say anything about the form of the density under the alternative. The results concern most powerful similar or unbiased tests, and under certain conditions the form of these tests for the particular family of densities studied, is given in Section 3. In Section 4 the theory in Section 3 is applied to the problem of testing serial correlation (not circular) in a first order autoregressive sequence. It is found that the usual tests are nearly UMP invariant. In Section 5 the problem of testing the value of the ratio of variances in the one-way classification variance components model is considered. Some numerical results are given for the power functions of the maximin test, the locally most powerful test and the standard $F$-test. The results indicate that the standard $F$-test performs well compared with the other tests.