Hypotheses testing by convex optimization

We discuss a general approach to hypothesis testing. The main “building block” of the proposed construction is a test for a pair of hypotheses in the situation where each particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypothesis. This test, under appropriate assumptions, is provably nearly optimal and is yielded by a solution to a convex optimization problem, so that the construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general testing problems encompassing several classical statistical settings.