Adaptive goodness-of-fit tests based on signed ranks

TL;DR

This paper introduces a new multiscale signed rank test for nonparametric regression that is distribution-free, adaptive to unknown smoothness, and asymptotically optimal, providing a robust method for testing regression functions.

Contribution

It proposes a novel multiscale signed rank statistic for nonparametric regression that is distribution-free, adaptive, and asymptotically optimal under various conditions.

Findings

Test is distribution-free and exact for finite samples.

Test adapts to unknown smoothness of the regression function.

Achieves asymptotic optimality and near-minimax efficiency.

Abstract

Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l=0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against H\"{o}lder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of H\"{o}lder classes simultaneously.