Discussion of "Hypothesis testing by convex optimization"

It is my pleasure to congratulate the authors for this insightful and innovative piece of work. Goldenshluger, Juditsky and Nemirovski (henceforth GJN) have proposed a unifying view on a broad family of hypothesis testing problems. They consider a composite hypothesis testing problem where the goal is to identify which of two convex sets do the parameters of the distribution lie in. Remarkably, the authors provide a set of conditions under which this composite testing problem boils down to a simple test between two appropriately chosen parameters, one from each set. The authors establish near-optimality guarantees for their procedure under favorable conditions. Furthermore, the underlying computation for obtaining the test can be cast as a convex optimization problem of a form for which efficient solvers are often available. This leads to a rather comprehensive solution, in both statistical and computational terms, of a class of hypothesis testing problems. © 2015, Institute of Mathematical Statistics. All rights reserved.