On Consistent Hypothesis Testing

Abstract: This paper explores conditions of existence of different types of consistent tests. New links of these types of consistency are also established. The existence of discernible (strong consistent) tests follows from the existence of pointwise consistent tests. If there are consistent tests, then the set of alternatives can be represented as countable union of nested subsets such that there are uniformly consistent tests for the hypothesis and each of this subset of alternatives. Implementing these results we explore both sufficient conditions and necessary conditions for existence of consistent, pointwise consistent, uniformly consistent and discernible tests for hypothesis testing on a probability measure of independent sample, on a mean measure of Poisson process, on a solution of linear ill-posed problems in Gaussian noise, on a solution of deconvolution problem and for the problem of signal detection in Gaussian white noise. In the last three cases the necessary conditions and sufficient conditions coincide.