Minimum distance regression model checking with Berkson measurement errors

TL;DR

This paper introduces a new class of goodness-of-fit tests for regression models with Berkson measurement errors, using minimum distance methods, with proven asymptotic properties and demonstrated finite sample performance.

Contribution

It develops the first lack-of-fit testing procedure for regression models with Berkson measurement errors, including asymptotic theory and simulation validation.

Findings

Asymptotic normality of test statistics under null hypothesis

Consistency of tests against fixed alternatives

Good finite sample performance in simulations

Abstract

Lack-of-fit testing of a regression model with Berkson measurement error has not been discussed in the literature to date. To fill this void, we propose a class of tests based on minimized integrated square distances between a nonparametric regression function estimator and the parametric model being fitted. We prove asymptotic normality of these test statistics under the null hypothesis and that of the corresponding minimum distance estimators under minimal conditions on the model being fitted. We also prove consistency of the proposed tests against a class of fixed alternatives and obtain their asymptotic power against a class of local alternatives orthogonal to the null hypothesis. These latter results are new even when there is no measurement error. A simulation that is included shows very desirable finite sample behavior of the proposed inference procedures.