Predictors with measurement error in mixtures of polynomial regressions

Xiaoqiong Fang, Andy W. Chen, Derek S. Young

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)373-401
Number of pages29
JournalComputational Statistics
Issue number1
StatePublished - Mar 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.


  • Bootstrap
  • Finite mixture models
  • GEM algorithm
  • Model selection
  • Regression calibration
  • Surrogate data

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Mathematics


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