Testing the suitability of polynomial models in errors-in-variables problems. (English) Zbl 1129.62042

Hall, Peter; Ma, Yanyuan

Ann. Stat. 35, No. 6, 2620-2638 (2007).

Summary: 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. 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 the experimental error, and, in particular, we depart substantially from conventional parametric models for errors-in-variables problems.

MSC:

62G10	Nonparametric hypothesis testing
62G08	Nonparametric regression and quantile regression
62G09	Nonparametric statistical resampling methods
62G20	Asymptotic properties of nonparametric inference

Keywords:

bandwidth; bootstrap; deconvolution; distribution estimation; ill-posed problem; kernel methods; measurement error; moment-matching bootstrap; smoothing; regularization; wild bootstrap

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