On Consistent Hypothesis Testing

Abstract

We study natural links between various types of consistency: usual consistency, strong consistency, uniform consistency, and pointwise consistency. On the base of these results, we provide both sufficient conditions and necessary conditions for the existence of various types of consistent tests for a wide spectrum of problems of hypothesis testing which arise in statistics: on a probability measure of an independent sample, on a mean measure of a Poisson process, on a solution of an ill-posed linear problem in a Gaussian noise, on a solution of the deconvolution problem, for signal detection in a Gaussian white noise. In the last three cases, the necessary and sufficient conditions coincide.