IDEAS home Printed from https://ideas.repec.org/a/bla/scjsta/v30y2003i1p45-59.html
   My bibliography  Save this article

Penalized Maximum Likelihood Estimator for Normal Mixtures

Author

Listed:
  • GABRIELA CIUPERCA
  • ANDREA RIDOLFI
  • JÉRÔME IDIER

Abstract

The estimation of the parameters of a mixture of Gaussian densities is considered, within the framework of maximum likelihood. Due to unboundedness of the likelihood function, the maximum likelihood estimator fails to exist. We adopt a solution to likelihood function degeneracy which consists in penalizing the likelihood function. The resulting penalized likelihood function is then bounded over the parameter space and the existence of the penalized maximum likelihood estimator is granted. As original contribution we provide asymptotic properties, and in particular a consistency proof, for the penalized maximum likelihood estimator. Numerical examples are provided in the finite data case, showing the performances of the penalized estimator compared to the standard one.

Suggested Citation

  • Gabriela Ciuperca & Andrea Ridolfi & Jérôme Idier, 2003. "Penalized Maximum Likelihood Estimator for Normal Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 45-59, March.
    • Handle: RePEc:bla:scjsta:v:30:y:2003:i:1:p:45-59
    DOI: 10.1111/1467-9469.00317
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/1467-9469.00317
    Download Restriction: no
    File URL: https://libkey.io/10.1111/1467-9469.00317?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Salem Reyen & John Miller & Edward Wegman, 2009. "Separating a mixture of two normals with proportional covariances," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 297-314, November.
    2. Seo, Byungtae & Lindsay, Bruce G., 2010. "A computational strategy for doubly smoothed MLE exemplified in the normal mixture model," Computational Statistics & Data Analysis, Elsevier, vol. 54(8), pages 1930-1941, August.
    3. Tin Lok James Ng & Thomas Brendan Murphy, 2021. "Model-based Clustering of Count Processes," Journal of Classification, Springer;The Classification Society, vol. 38(2), pages 188-211, July.
    4. Roberto Rocci & Stefano Antonio Gattone & Roberto Di Mari, 2018. "A data driven equivariant approach to constrained Gaussian mixture modeling," 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. 12(2), pages 235-260, June.
    5. Mingxing He & Jiahua Chen, 2022. "Consistency of the MLE under a two-parameter Gamma mixture model with a structural shape parameter," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(8), pages 951-975, November.
    6. Hai Shu & Bin Nan & Robert Koeppe, 2015. "Multiple testing for neuroimaging via hidden Markov random field," Biometrics, The International Biometric Society, vol. 71(3), pages 741-750, September.
    7. Yeojin Chung & Sophia Rabe-Hesketh & Vincent Dorie & Andrew Gelman & Jingchen Liu, 2013. "A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models," Psychometrika, Springer;The Psychometric Society, vol. 78(4), pages 685-709, October.
    8. Kim, Daeyoung & Seo, Byungtae, 2014. "Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 100-120.
    9. Yeojin Chung & Andrew Gelman & Sophia Rabe-Hesketh & Jingchen Liu & Vincent Dorie, 2015. "Weakly Informative Prior for Point Estimation of Covariance Matrices in Hierarchical Models," Journal of Educational and Behavioral Statistics, , vol. 40(2), pages 136-157, April.
    10. Nicolas Depraetere & Martina Vandebroek, 2014. "Order selection in finite mixtures of linear regressions," Statistical Papers, Springer, vol. 55(3), pages 871-911, August.
    11. Seo, Byungtae & Kim, Daeyoung, 2012. "Root selection in normal mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2454-2470.
    12. Hentati Rania & Prigent Jean-Luc, 2011. "On the maximization of financial performance measures within mixture models," Statistics & Risk Modeling, De Gruyter, vol. 28(1), pages 63-80, March.
    13. Mingxing He & Jiahua Chen, 2022. "Strong consistency of the MLE under two-parameter Gamma mixture models with a structural scale parameter," 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. 16(1), pages 125-154, March.
    14. Luis Angel García-Escudero & Alfonso Gordaliza & Francesca Greselin & Salvatore Ingrassia & Agustín Mayo-Iscar, 2018. "Eigenvalues and constraints in mixture modeling: geometric and computational issues," 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. 12(2), pages 203-233, June.
    15. Maddalena Cavicchioli, 2016. "Statistical Analysis Of Mixture Vector Autoregressive Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1192-1213, December.
    16. Alexandrovich, Grigory, 2014. "A note on the article ‘Inference for multivariate normal mixtures’ by J. Chen and X. Tan," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 245-248.
    17. Tin Lok James Ng, 2023. "Penalized maximum likelihood estimator for mixture of von Mises–Fisher distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(2), pages 181-203, February.
    18. Wenguang Sun & T. Tony Cai, 2009. "Large‐scale multiple testing under dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 393-424, April.
    19. Bia, Michela & Flores-Lagunes, Alfonso & Mercatanti, Andrea, 2018. "Evaluation of Language Training Programs in Luxembourg Using Principal Stratification," IZA Discussion Papers 11973, Institute of Labor Economics (IZA).
    20. Kentaro Tanaka, 2009. "Strong Consistency of the Maximum Likelihood Estimator for Finite Mixtures of Location–Scale Distributions When Penalty is Imposed on the Ratios of the Scale Parameters," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 171-184, March.
    21. Yin, Cuihong & Sheldon Lin, X. & Huang, Rongtan & Yuan, Haili, 2019. "On the consistency of penalized MLEs for Erlang mixtures," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 12-20.
    22. Ingrassia, Salvatore & Rocci, Roberto, 2007. "Constrained monotone EM algorithms for finite mixture of multivariate Gaussians," Computational Statistics & Data Analysis, Elsevier, vol. 51(11), pages 5339-5351, July.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:scjsta:v:30:y:2003:i:1:p:45-59. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0303-6898 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.