Hypothesis Testing when the Sample Size is Treated as a

SUMMARY In the Neyman-Pearson theory of hypothesis testing it is customary to calculate significance levels and power functions on the assumption that the sample size is fixed. The main purpose of this paper is to propagate the view that the restriction of the theory to reference sets in which the sample size is constant is neither necessary nor desirable. The paper falls into two parts. In the first we consider the test of a simple null hypothesis versus a simple alternative. In this case it is pointed out that the optimum test requires the rejection of the null hypothesis if the likelihood ratio is less than a constant, K, which is independent of n, the sample size. In this respect it is identical with Bayesian and likelihood methods of dealing with the same problem. Two methods for determining a suitable value for K in the absence of knowledge about the distribution of sample size are proposed. These ideas are extended, in the second part of the paper, to the case of a simple null hypothesis and a composite alternative. Since no uniformly most powerful test exists, a variety of tests suggested by Bayesian and likelihood considerations are compared from a frequentist point of view. This work is exploratory, being based on numerical calculations, but provides a partial justification for treating sample size as if it were fixed.