Testing the suitability of polynomial models in errors-in-variables problems

TL;DR

This paper develops novel methods for testing the goodness of fit of polynomial models in errors-in-variables problems, especially when covariate errors are present and traditional bootstrap methods are not applicable.

Contribution

It introduces a new approach using deconvolution and wild bootstrap techniques to assess polynomial model fit under covariate measurement errors, without relying on parametric error assumptions.

Findings

Effective testing methods for polynomial models with covariate errors.

Approach applicable when covariate error distribution is known or estimated.

No reliance on parametric assumptions for experimental errors.

Abstract

A low-degree polynomial model for a response curve is used commonly in practice. It generally incorporates a linear or quadratic function of the covariate. In this paper we suggest methods for testing the goodness of fit of a general polynomial model when there are errors in the covariates. There, the true covariates are not directly observed, and conventional bootstrap methods for testing are not applicable. We develop a new approach, in which deconvolution methods are used to estimate the distribution of the covariates under the null hypothesis, and a ``wild'' or moment-matching bootstrap argument is employed to estimate the distribution of the experimental errors (distinct from the distribution of the errors in covariates). Most of our attention is directed at the case where the distribution of the errors in covariates is known, although we also discuss methods for estimation and testing when the covariate error distribution is estimated. No assumptions are made about the distribution of experimental error, and, in particular, we depart substantially from conventional parametric models for errors-in-variables problems.