On Consistent Hypothesis Testing

We explore conditions of existence of consistent, uniformly consistent and discernible (strong consistent) tests. We establish that the existence of discernible tests follows from the existence of pointwise consistent tests. We show that, 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 these subsets of alternatives. Implementing these results we explore both sufficient conditions and necessary conditions for existence of 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.