On consistency and inconsistency of nonparametric tests

TL;DR

This paper investigates the conditions under which various nonparametric tests, including chi-squared and kernel-based tests, are consistent or inconsistent, providing insights into their effectiveness for different alternative sequences.

Contribution

It establishes necessary and sufficient conditions for the consistency of multiple nonparametric tests in terms of alternative sequences and introduces interpretations of these conditions.

Findings

Necessary and sufficient conditions for test consistency.

Consistency depends on the structure of alternative sequences.

Compactness of the alternative set is necessary for signal detection in Gaussian noise.

Abstract

For $\chi^2-$tests with increasing number of cells, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients, we find necessary and sufficient conditions of consistency and inconsistency for sequences of alternatives having a given rate of convergence to hypothesis in $\mathbb{L}_2$-norm. We provide transparent interpretations of these conditions allowing to understand the structure of such consistent sequences. For problem of signal detection in Gaussian white noise we show that, if set of alternatives is bounded closed center-symmetric convex set $U$ with deleted "small" $\mathbb{L}_2$ -- ball, then compactness of set $U$ is necessary condition for existence of consistent tests.