The Annals of Statistics

Goodness-of-fit testing of error distribution in linear measurement error models

Hira L. Koul, Weixing Song, and Xiaoqing Zhu

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This paper investigates a class of goodness-of-fit tests for fitting an error density in linear regression models with measurement error in covariates. Each test statistic is the integrated square difference between the deconvolution kernel density estimator of the regression model error density and a smoothed version of the null error density, an analog of the so-called Bickel and Rosenblatt test statistic. The asymptotic null distributions of the proposed test statistics are derived for both the ordinary smooth and super smooth cases. The asymptotic power behavior of the proposed tests against a fixed alternative and a class of local nonparametric alternatives for both cases is also described. The finite sample performance of the proposed test is evaluated by a simulation study. The simulation study shows some superiority of the proposed test over some other tests. Finally, a real data is used to illustrate the proposed test.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 2479-2510.

Received: September 2015
Revised: July 2017
First available in Project Euclid: 17 August 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Deconvolution density estimators $L_{2}$-distance tests


Koul, Hira L.; Song, Weixing; Zhu, Xiaoqing. Goodness-of-fit testing of error distribution in linear measurement error models. Ann. Statist. 46 (2018), no. 5, 2479--2510. doi:10.1214/17-AOS1627.

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Supplemental materials

  • Some simulation results of GOF tests in measurement error models. This supplement contains some additional simulation results comparing the test proposed in this paper with some other tests and a bandwidth sensitivity analysis.