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Predictors with measurement error in mixtures of polynomial regressions

Author

Listed:
  • Xiaoqiong Fang

    (Corporate & Investment Bank, J.P. Morgan)

  • Andy W. Chen

    (School of Business, Government, and Economics, Seattle Pacific University)

  • Derek S. Young

    (University of Kentucky)

Abstract

There has been a substantial body of research on mixtures-of-regressions models that has developed over the past 20 years. While much of the recent literature has focused on flexible mixtures-of-regressions models, there is still considerable utility for imposing structure on the mixture components through fully parametric models. One feature of the data that is scantly addressed in mixtures of regressions is the presence of measurement error in the predictors. The limited existing research on this topic concerns the case where classical measurement error is added to the classic mixtures-of-linear-regressions model. In this paper, we consider the setting of mixtures of polynomial regressions where the predictors are subject to classical measurement error. Moreover, each component is allowed to have a different degree for the polynomial structure. We utilize a generalized expectation-maximization algorithm for performing maximum likelihood estimation. For estimating standard errors, we extend a semiparametric bootstrap routine that has been employed for mixtures of linear regressions without measurement error in the predictors. Numeric work, for practical reasons identified, is limited to estimating two-component models. We consider a likelihood ratio test for determining if there is a higher-degree polynomial term in one of the components. Model selection criteria are also highlighted as a way for determining an appropriate model. A simulation study and an application to the classic nitric oxide emissions data are provided.

Suggested Citation

  • Xiaoqiong Fang & Andy W. Chen & Derek S. Young, 2023. "Predictors with measurement error in mixtures of polynomial regressions," Computational Statistics, Springer, vol. 38(1), pages 373-401, March.
    • Handle: RePEc:spr:compst:v:38:y:2023:i:1:d:10.1007_s00180-022-01232-5
    DOI: 10.1007/s00180-022-01232-5
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    References listed on IDEAS

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    1. De Veaux, Richard D., 1989. "Mixtures of linear regressions," Computational Statistics & Data Analysis, Elsevier, vol. 8(3), pages 227-245, November.
    2. Salvatore Ingrassia & Antonio Punzo & Giorgio Vittadini & Simona Minotti, 2015. "Erratum to: The Generalized Linear Mixed Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 32(2), pages 327-355, July.
    3. T. Rolf Turner, 2000. "Estimating the propagation rate of a viral infection of potato plants via mixtures of regressions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 49(3), pages 371-384.
    4. Hamparsum Bozdogan, 1987. "Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 345-370, September.
    5. Elizabeth A. Sugar & Ching-Yun Wang & Ross L. Prentice, 2007. "Logistic Regression with Exposure Biomarkers and Flexible Measurement Error," Biometrics, The International Biometric Society, vol. 63(1), pages 143-151, March.
    6. Matthew Stephens, 2000. "Dealing with label switching in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 795-809.
    7. David Hunter & Derek Young, 2012. "Semiparametric mixtures of regressions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(1), pages 19-38.
    8. Weixin Yao & Weixing Song, 2015. "Mixtures of Linear Regression with Measurement Errors," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(8), pages 1602-1614, April.
    9. Pierre Vandekerkhove, 2013. "Estimation of a semiparametric mixture of regressions model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(1), pages 181-208, March.
    10. Peter Lenk & Wayne DeSarbo, 2000. "Bayesian inference for finite mixtures of generalized linear models with random effects," Psychometrika, Springer;The Psychometric Society, vol. 65(1), pages 93-119, March.
    11. Sylvia Richardson & Laurent Leblond & Isabelle Jaussent & Peter J. Green, 2002. "Mixture models in measurement error problems, with reference to epidemiological studies," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 165(3), pages 549-566, October.
    12. Ingrassia, Salvatore & Minotti, Simona C. & Punzo, Antonio, 2014. "Model-based clustering via linear cluster-weighted models," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 159-182.
    13. Matthew Blackwell & James Honaker & Gary King, 2017. "A Unified Approach to Measurement Error and Missing Data: Overview and Applications," Sociological Methods & Research, , vol. 46(3), pages 303-341, August.
    14. Salvatore Ingrassia & Antonio Punzo & Giorgio Vittadini & Simona Minotti, 2015. "The Generalized Linear Mixed Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 32(1), pages 85-113, April.
    15. Michel Wedel & Wayne DeSarbo, 1995. "A mixture likelihood approach for generalized linear models," Journal of Classification, Springer;The Classification Society, vol. 12(1), pages 21-55, March.
    16. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "The bivariate Sinh-Elliptical distribution with applications to Birnbaum–Saunders distribution and associated regression and measurement error models," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 1-16.
    17. Salvatore Ingrassia & Simona Minotti & Giorgio Vittadini, 2012. "Local Statistical Modeling via a Cluster-Weighted Approach with Elliptical Distributions," Journal of Classification, Springer;The Classification Society, vol. 29(3), pages 363-401, October.
    18. Rohit Kumar Patra & Bodhisattva Sen, 2016. "Estimation of a two-component mixture model with applications to multiple testing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 869-893, September.
    19. G. J. McLachlan, 1987. "On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(3), pages 318-324, November.
    20. Raymond J. Carroll & Kathryn Roeder & Larry Wasserman, 1999. "Flexible Parametric Measurement Error Models," Biometrics, The International Biometric Society, vol. 55(1), pages 44-54, March.
    21. Clifford M. Hurvich & Jeffrey S. Simonoff & Chih‐Ling Tsai, 1998. "Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 271-293.
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