The Annals of Statistics

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

Peter Hall and Yanyuan Ma

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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.

Article information

Ann. Statist., Volume 35, Number 6 (2007), 2620-2638.

First available in Project Euclid: 22 January 2008

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Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G09: Resampling methods 62G10: Hypothesis testing 62G20: Asymptotic properties

Bandwidth bootstrap deconvolution distribution estimation hypothesis testing ill-posed problem kernel methods measurement error moment-matching bootstrap smoothing regularization wild bootstrap


Hall, Peter; Ma, Yanyuan. Testing the suitability of polynomial models in errors-in-variables problems. Ann. Statist. 35 (2007), no. 6, 2620--2638. doi:10.1214/009053607000000361.

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