Efficient nonparametric testing by functional estimation

Abstract

Many nonparametric tests admit improvement by identifying a functional on a set of probability measures ℱ, of which the test statistic is an estimator. We call such a functional a gauge for the problem if it induces the partition of ℱ into null and alternative and enjoys certain invariance properties. Two nonparametric testing problems are explored here: a dependency problem and an equidistribution problem. In each a dual smoothing problem is posed and optimally solved in the estimation framework, and a corresponding testing procedure gives a consistency rate improvement over the original test.