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Non-parametric identification and estimation of the number of components in multivariate mixtures

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  • Hiroyuki Kasahara
  • Katsumi Shimotsu

Abstract

type="main" xml:id="rssb12022-abs-0001"> We analyse 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 non-parametrically 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 estimate a lower bound on the number of components consistently.

Suggested Citation

  • Hiroyuki Kasahara & Katsumi Shimotsu, 2014. "Non-parametric identification and estimation of the number of components in multivariate mixtures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(1), pages 97-111, January.
    • Handle: RePEc:bla:jorssb:v:76:y:2014:i:1:p:97-111
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    File URL: http://hdl.handle.net/10.1111/rssb.2013.76.issue-1
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    Cited by:

    1. Konstantin T. Matchev & Prasanth Shyamsundar, 2020. "InClass Nets: Independent Classifier Networks for Nonparametric Estimation of Conditional Independence Mixture Models and Unsupervised Classification," Papers 2009.00131, arXiv.org.
    2. Hu, Yingyao & Xin, Yi, 2024. "Identification and estimation of dynamic structural models with unobserved choices," Journal of Econometrics, Elsevier, vol. 242(2).
    3. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Nonparametric spectral-based estimation of latent structures," CeMMAP working papers CWP18/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Paul Schrimpf & Michio Suzuki & Hiroyuki Kasahara, 2015. "Identification and Estimation of Production Function with Unobserved Heterogeneity," 2015 Meeting Papers 924, Society for Economic Dynamics.
    5. repec:spo:wpmain:info:hdl:2441/etefo8s8r89oamhnhiclqr530 is not listed on IDEAS
    6. Johannes F. Jörg & Catherine Cleophas, 2022. "Nonparametric estimation of customer segments from censored sales panel data," Journal of Revenue and Pricing Management, Palgrave Macmillan, vol. 21(4), pages 393-417, August.
    7. Yu Hao & Hiroyuki Kasahara, 2025. "Estimating the Number of Components in Panel Data Finite Mixture Regression Models with an Application to Production Function Heterogeneity," Papers 2506.09666, arXiv.org.
    8. Erhao Xie, 2018. "Inference in Games Without Nash Equilibrium: An Application to Restaurants, Competition in Opening Hours," Staff Working Papers 18-60, Bank of Canada.
    9. 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.
    10. Marie du Roy de Chaumaray & Matthieu Marbac, 2023. "Clustering data with non-ignorable missingness using semi-parametric mixture models assuming independence within components," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(4), pages 1081-1122, December.
    11. 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.
    12. 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.
    13. Yu Hao & Hiroyuki Kasahara, 2022. "Testing the Number of Components in Finite Mixture Normal Regression Model with Panel Data," Papers 2210.02824, arXiv.org, revised Jun 2023.
    14. 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.
    15. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Estimating Multivariate Latent-Structure Models," Sciences Po Economics Publications (main) hal-01097135, HAL.
    16. Michael Levine & Gildas Mazo, 2024. "A smoothed semiparametric likelihood for estimation of nonparametric finite mixture models with a copula-based dependence structure," Computational Statistics, Springer, vol. 39(4), pages 1825-1846, June.
    17. Xu, Ke-Li, 2018. "A semi-nonparametric estimator of regression discontinuity design with discrete duration outcomes," Journal of Econometrics, Elsevier, vol. 206(1), pages 258-278.
    18. Qihui Chen & Zheng Fang, 2018. "Improved Inference on the Rank of a Matrix," Papers 1812.02337, arXiv.org, revised Mar 2019.
    19. 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.
    20. Krasnokutskaya, Elena & Song, Kyungchul & Tang, Xun, 2022. "Estimating unobserved individual heterogeneity using pairwise comparisons," Journal of Econometrics, Elsevier, vol. 226(2), pages 477-497.
    21. Bagkavos, Dimitrios & Patil, Prakash N., 2023. "Goodness-of-fit testing for normal mixture densities," Computational Statistics & Data Analysis, Elsevier, vol. 188(C).
    22. Áureo de Paula & Xun Tang, 2020. "Testable implications of multiple equilibria in discrete games with correlated types," CeMMAP working papers CWP56/20, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    23. 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.
    24. Hiroaki Masuhara, 2019. "Identifying finite mixture models in the presence of moment-generating function: application in medical care using a zero-inflated binomial model," Economics Bulletin, AccessEcon, vol. 39(2), pages 1529-1537.

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