Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression

TL;DR

This paper develops minimax testing procedures for a specific quadratic functional in nonparametric multivariate regression, identifying optimal rates and separation constants for different functional regimes and applying results to norm equality testing.

Contribution

It provides the first sharp minimax rates and optimal testing procedures for quadratic functionals under various smoothness classes, including anisotropic Sobolev spaces.

Findings

Minimax rates are established for diagonal and nonnegative quadratic functionals.

Distinct 'regular' and 'irregular' regimes are characterized with different rates.

Application to testing equality of norms in noisy environments demonstrates practical relevance.

Abstract

We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$ satisfies the constraint $Q[f]=0$, while the alternative corresponds to the functions for which $Q[f]$ is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: "regular" and "irregular". In the "regular" case, the minimax rate is equal to $n^{-1/4}$ while in the "irregular" case, the rate depends on the smoothness class and is slower than in the "regular" case. We apply this to the issue of testing the equality of norms of two functions observed in noisy environments.