On maxispaces of nonparametric tests

TL;DR

This paper introduces the concepts of maxisets and maxispace in nonparametric hypothesis testing, characterizing the optimal sets of alternatives for various tests and identifying asymptotically minimax tests.

Contribution

It defines maxisets and maxispaces for nonparametric tests and characterizes their properties across multiple test types, advancing understanding of optimal testing strategies.

Findings

Maxisets of chi-squared and Cramer-von Mises tests are characterized.

Any consistent alternative can be decomposed into parts within and orthogonal to the maxiset.

Asymptotically minimax tests are identified using maxisets with small L2-balls removed.

Abstract

For the problems of nonparametric hypothesis testing we introduce the notion of maxisets and maxispace. We point out the maxisets of $\chi^2-$tests, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients. For these tests we show that, if sequence of alternatives having given rates of convergence to hypothesis is consistent, then each altehrnative can be broken down into the sum of two parts: a function belonging to maxiset and orthogonal function. Sequence of functions belonging to maxiset is consistent sequence of alternatives. We point out asymptotically minimax tests if sets of alternatives are maxiset with deleted "small" $\mathbb{L}_2$-balls.