### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 2620-2638 |

Number of pages | 19 |

Journal | Annals of Statistics |

Volume | 35 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Statistics*,

*35*(6), 2620-2638. https://doi.org/10.1214/009053607000000361

}

*Annals of Statistics*, vol. 35, no. 6, pp. 2620-2638. https://doi.org/10.1214/009053607000000361

**Testing the suitability of polynomial models in errors-in-variables problems.** / Hall, Peter; Ma, Yanyuan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Testing the suitability of polynomial models in errors-in-variables problems

AU - Hall, Peter

AU - Ma, Yanyuan

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=50849118145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50849118145&partnerID=8YFLogxK

U2 - 10.1214/009053607000000361

DO - 10.1214/009053607000000361

M3 - Article

VL - 35

SP - 2620

EP - 2638

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 6

ER -