A projection-based adaptive-to-model test for regressions

TL;DR

This paper introduces a projection-based adaptive-to-model test for regressions that effectively handles multiple covariates, providing an omnibus, distribution-free testing procedure that adapts to the underlying model structure.

Contribution

It proposes a novel projection-based adaptive-to-model testing approach that achieves distribution-free null distribution and adapts to model structure, improving over existing methods.

Findings

The test is asymptotically distribution-free under the null hypothesis.

It can automatically adapt to the true model structure, making it omnibus.

Simulation studies and real data analysis demonstrate the effectiveness of the method.

Abstract

A longstanding problem of existing empirical process-based tests for regressions is that when the number of covariates is greater than one, they either have no tractable limiting null distributions or are not omnibus. To attack this problem, we in this paper propose a projection-based adaptive-to-model approach. When the hypothetical model is parametric single-index, the method can fully utilize the dimension reduction model structure under the null hypothesis as if the covariate were one-dimensional such that the martingale transformation-based test can be asymptotically distribution-free. Further, the test can automatically adapt to the underlying model structure such that the test can be omnibus and thus detect alternative models distinct from the hypothetical model at the fastest possible rate in hypothesis testing. The method is examined through simulation studied and is illustrated by a real data analysis.