Nonparametric testing by convex optimization

We discuss a general approach to handling a class of nonparametric detection problems when the null and each particular alternative hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact. Our central result is a test for a pair of hypotheses of the outlined type which, under appropriate assumptions, is provably nearly optimal. The test is yielded by a solution to a convex programming problem, and, as a result, the proposed construction admits a computationally efficient implementation. We show how our approach can be applied to a rather general detection problem encompassing several classical statistical settings such as detection of abrupt signal changes, cusp detection and multi-sensor detection. [Joint work with Alexander Goldenshluger and Arkadi Nemirovski]