Robust tests for equality of regression curves based on characteristic functions

TL;DR

This paper introduces a robust testing method for equality of regression functions using characteristic functions and residuals from a robust smoother, outperforming classical methods especially with finite samples.

Contribution

It proposes a new test statistic combining characteristic functions and robust residuals, improving reliability over traditional kernel-based tests in nonparametric regression.

Findings

The proposed test has better finite sample performance than classical methods.

Monte Carlo simulations demonstrate robustness against atypical responses.

Application to real data shows sensitivity to bandwidth choice.

Abstract

This paper focuses on the problem of testing the null hypothesis that the regression functions of several populations are equal under a general nonparametric homoscedastic regression model. It is well known that linear kernel regression estimators are sensitive to atypical responses. These distorted estimates will influence the test statistic constructed from them so the conclusions obtained when testing equality of several regression functions may also be affected. In recent years, the use of testing procedures based on empirical characteristic functions has shown good practical properties. For that reason, to provide more reliable inferences, we construct a test statistic that combines characteristic functions and residuals obtained from a robust smoother under the null hypothesis. The asymptotic distribution of the test statistic is studied under the null hypothesis and under root$-n$ contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behaviour of the proposed test with the classical one obtained using local averages. The reported numerical experiments show the advantage of the proposed methodology over the one based on Nadaraya-Watson estimators for finite samples. An illustration to a real data set is also provided and enables to investigate the sensitivity of the $p-$value to the bandwidth selection.