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Nonparametric Identification and Estimation of the Number of Components in Multivariate Mixtures

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  • Hiroyuki Kasahara

    (Department of Economics, University of British Columbia)

  • Katsumi Shimotsu

    (Faculty of Economics, University of Tokyo)

Abstract

This article analyzes the identifiability of the number of components in k-variate, M-component finite mixture models in which each component distribution has independent marginals, including models in latent class analysis. Without making parametric assumptions on the component distributions, we investigate how one can identify the number of components from the distribution function of the observed data. When k>=2, a lower bound on the number of components (M) is nonparametrically identifiable from the rank of a matrix constructed from the distribution function of the observed variables. Building on this identification condition, we develop a procedure to consistently estimate a lower bound on the number of components.

Suggested Citation

  • Hiroyuki Kasahara & Katsumi Shimotsu, 2012. "Nonparametric Identification and Estimation of the Number of Components in Multivariate Mixtures," CIRJE F-Series CIRJE-F-866, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2012cf866
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    References listed on IDEAS

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    1. Andrews, Donald W. K., 1987. "Asymptotic Results for Generalized Wald Tests," Econometric Theory, Cambridge University Press, vol. 3(03), pages 348-358, June.
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    3. Woo, Mi-Ja & Sriram, T.N., 2006. "Robust Estimation of Mixture Complexity," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1475-1486, December.
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    6. Peter Hall & Amnon Neeman & Reza Pakyari & Ryan Elmore, 2005. "Nonparametric inference in multivariate mixtures," Biometrika, Biometrika Trust, vol. 92(3), pages 667-678, September.
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    8. Robert Mislevy, 1984. "Estimating latent distributions," Psychometrika, Springer;The Psychometric Society, vol. 49(3), pages 359-381, September.
    9. Hiroyuki Kasahara & Katsumi Shimotsu, 2009. "Nonparametric Identification of Finite Mixture Models of Dynamic Discrete Choices," Econometrica, Econometric Society, vol. 77(1), pages 135-175, January.
    10. Lutkepohl, Helmut & Burda, Maike M., 1997. "Modified Wald tests under nonregular conditions," Journal of Econometrics, Elsevier, vol. 78(2), pages 315-332, June.
    11. I. R. Cruz-Medina & T. P. Hettmansperger & H. Thomas, 2004. "Semiparametric mixture models and repeated measures: the multinomial cut point model," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 53(3), pages 463-474.
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    Cited by:

    1. Jean-Marc Robin & Stéphane Bonhomme & Koen Jochmans, 2014. "Estimating Multivariate Latent-Structure Models," Sciences Po Economics Discussion Papers 2014-18, Sciences Po Departement of Economics.
    2. repec:eee:econom:v:201:y:2017:i:2:p:237-248 is not listed on IDEAS
    3. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2016. "Non-parametric estimation of finite mixtures from repeated measurements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 211-229, January.
    4. Bonhomme, Stéphane & Jochmans, Koen & Robin, Jean-Marc, 2017. "Nonparametric estimation of non-exchangeable latent-variable models," Journal of Econometrics, Elsevier, vol. 201(2), pages 237-248.
    5. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2017. "Nonparametric estimation of non-exchangeable latent-variable models," Sciences Po publications info:hdl:2441/4m4fqk908d9, Sciences Po.
    6. David Balan & Patrick DeGraba & Francine Lafontaine & Patrick McAlvanah & Devesh Raval & David Schmidt, 2015. "Economics at the FTC: Fraud, Mergers and Exclusion," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 47(4), pages 371-398, December.

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