Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach

TL;DR

This paper develops asymptotically distribution-free goodness-of-fit tests for error distributions in nonparametric regression, improving power by using a martingale transform of residuals that accounts for covariates.

Contribution

It introduces a new distribution-free testing approach based on martingale transforms applicable under broad conditions in nonparametric regression.

Findings

Tests have better power due to elimination of covariate-induced randomness.

Applicable under conditions of asymptotic uniform linearity of residual processes.

Method extends existing frameworks to broader nonparametric models.

Abstract

This paper discusses asymptotically distribution free tests for the classical goodness-of-fit hypothesis of an error distribution in nonparametric regression models. These tests are based on the same martingale transform of the residual empirical process as used in the one sample location model. This transformation eliminates extra randomization due to covariates but not due the errors, which is intrinsically present in the estimators of the regression function. Thus, tests based on the transformed process have, generally, better power. The results of this paper are applicable as soon as asymptotic uniform linearity of nonparametric residual empirical process is available. In particular they are applicable under the conditions stipulated in recent papers of Akritas and Van Keilegom and M\"uller, Schick and Wefelmeyer.