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Testing the Number of Components in Finite Mixture Models


  • Hiroyuki Kasahara

    (Department of Economics, University of British Columbia)

  • Katsumi Shimotsu

    (Faculty of Economics, University of Tokyo)


This paper considers likelihood-based testing of the null hypothesis of m0 components against the alternative of m 0+1 components in a finite mixture model. The number of components is an important parameter in the applications of finite mixture models. Still, testing the number of components has been a long-standing challenging problem because of its non-regularity. We develop a framework that facilitates the analysis of the likelihood function of finite mixture models and derive the asymptotic distribution of the likelihood ratio test statistic for testing the null hypothesis of m 0 components against the alternative of m 0+1 components. Furthermore, building on this framework, we propose a likelihood-based testing procedure of the number of components. The proposed test, extending the EM approach of Li, Chen and Marriott (2009), does not use a penalty term and is implementable even when the likelihood ratio test is difficult to implement because of non-regularity and computational complexity.

Suggested Citation

  • Hiroyuki Kasahara & Katsumi Shimotsu, 2012. "Testing the Number of Components in Finite Mixture Models," CIRJE F-Series CIRJE-F-867, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2012cf867

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    1. Stephen V. Cameron & James J. Heckman, 1998. "Life Cycle Schooling and Dynamic Selection Bias: Models and Evidence for Five Cohorts of American Males," Journal of Political Economy, University of Chicago Press, vol. 106(2), pages 262-333, April.
    2. Heckman, James & Singer, Burton, 1984. "A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data," Econometrica, Econometric Society, vol. 52(2), pages 271-320, March.
    3. Jiahua Chen & Pengfei Li & Yuejiao Fu, 2012. "Inference on the Order of a Normal Mixture," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1096-1105, September.
    4. Deb, Partha & Trivedi, Pravin K., 2002. "The structure of demand for health care: latent class versus two-part models," Journal of Health Economics, Elsevier, vol. 21(4), pages 601-625, July.
    5. Jin Seo Cho & Halbert White, 2007. "Testing for Regime Switching," Econometrica, Econometric Society, vol. 75(6), pages 1671-1720, November.
    6. P. Li & J. Chen & P. Marriott, 2009. "Non-finite Fisher information and homogeneity: an EM approach," Biometrika, Biometrika Trust, vol. 96(2), pages 411-426.
    7. Li, Pengfei & Chen, Jiahua, 2010. "Testing the Order of a Finite Mixture," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1084-1092.
    8. Hong-Tu Zhu & Heping Zhang, 2004. "Hypothesis testing in mixture regression models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 3-16.
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