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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aos/1534492842

Digital Object Identifier
doi:10.1214/17-AOS1627

Mathematical Reviews number (MathSciNet)
MR3845024

Zentralblatt MATH identifier
06964339

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1534492842


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References

  • Bachmann, D. and Dette, H. (2005). A note on the Bickel–Rosenblatt test in autoregressive time series. Statist. Probab. Lett. 74 221–234.
  • Battese, G. E., Fuller, W. and Hickman, R. D. (1976). Estimation of response variances from interview-reinterview surveys. J. Indian Soc. Agricultural Statist. 28 1–14.
  • Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • Butucea, C. (2004). Asymptotic normality of the integrated square error of a density estimator in the convolution model. SORT 28 9–25.
  • Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
  • Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective, 2nd ed. Monographs on Statistics and Applied Probability 105. Chapman & Hall/CRC, Boca Raton, FL.
  • Cheng, C.-L. and Van Ness, J. W. (1999). Statistical Regression with Measurement Error. Kendall’s Library of Statistics 6. Arnold, London.
  • Delaigle, A. and Hall, P. (2006). On optimal kernel choice for deconvolution. Statist. Probab. Lett. 76 1594–1602.
  • Fan, J. (1991). Asymptotic normality for deconvolution kernel density estimators. Sankhyā Ser. A 53 97–110.
  • Fan, J. (1992). Deconvolution with supersmooth distributions. Canad. J. Statist. 20 155–169.
  • Fuller, W. A. (1987). Measurement Error Models. Wiley, New York.
  • Gao, J. and Gijbels, I. (2008). Bandwidth selection in nonparametric kernel testing. J. Amer. Statist. Assoc. 103 1584–1594.
  • Holzmann, H., Bissantz, N. and Munk, A. (2007). Density testing in a contaminated sample. J. Multivariate Anal. 98 57–75.
  • Holzmann, H. and Boysen, L. (2006). Integrated square error asymptotics for supersmooth deconvolution. Scand. J. Stat. 33 849–860.
  • Hong, Y. and Lee, Y.-J. (2013). A loss function approach to model specification testing and its relative efficiency. Ann. Statist. 41 1166–1203.
  • Hušková, M. and Meintanis, S. G. (2007). Omnibus tests for the error distribution in the linear regression model. Statistics 41 363–376.
  • Khmaladze, E. V. and Koul, H. L. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist. 32 995–1034.
  • Khmaladze, E. V. and Koul, H. L. (2009). Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach. Ann. Statist. 37 3165–3185.
  • Koul, H. L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models. Lecture Notes in Statistics 166. Springer, New York. Second edition of Weighted Empiricals and Linear Models [Inst. Math. Statist., Hayward, CA, 1992; MR1218395].
  • Koul, H. L. and Mimoto, N. (2012). A goodness-of-fit test for GARCH innovation density. Metrika 75 127–149.
  • Koul, H. L. and Song, W. (2012). A class of goodness-of-fit tests in linear errors-in-variables model. J. SFdS 153 52–70.
  • Koul, H. L., Song, W. and Zhu, X. (2018). Supplement to “Goodness-of-fit testing of error distribution in linear measurement error models.” DOI:10.1214/17-AOS1627SUPP.
  • Laurent, B., Loubes, J.-M. and Marteau, C. (2011). Testing inverse problems: A direct or an indirect problem? J. Statist. Plann. Inference 141 1849–1861.
  • Lee, S. and Na, S. (2002). On the Bickel–Rosenblatt test for first-order autoregressive models. Statist. Probab. Lett. 56 23–35.
  • Loubes, J. M. and Marteau, C. (2014). Goodness-of-fit testing strategies from indirect observations. J. Nonparametr. Stat. 26 85–99.
  • Loynes, R. M. (1980). The empirical d.f. of residuals from generalized regression. Ann. Statist. 8 285–298.
  • Schennach, S. M. and Hu, Y. (2013). Nonparametric identification and semiparametric estimation of classical measurement error models without side information. J. Amer. Statist. Assoc. 108 177–186.
  • Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer, New York.
  • Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229–235.
  • Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
  • Van Es, A. and Uh, H.-W. (2004). Asymptotic normality of nonparametric kernel type deconvolution density estimators: Crossing the Cauchy boundary. J. Nonparametr. Stat. 16 261–277.

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.