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Root selection in normal mixture models

  • Seo, Byungtae
  • Kim, Daeyoung
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    Finite mixtures of normal distributions are attractive in identifying the underlying group structure in the data. However, it is a challenging task to do statistical inference in normal mixture models using the method of maximum likelihood, due to the unbounded likelihood and the existence of multiple roots to the likelihood equation including a so-called spurious root. In this article we propose a new likelihood-based method for selecting a statistically reasonable root when there exist multiple roots of the likelihood equation for a finite normal mixture model. We first prove that our proposed methodology can choose a root to the mixture likelihood equation with consistency. We then show, by simulation studies and real examples, that the proposed methods can greatly reduce the risk of choosing problematic roots that have the same features as spurious roots.

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    File URL: http://www.sciencedirect.com/science/article/pii/S016794731200062X
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    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 56 (2012)
    Issue (Month): 8 ()
    Pages: 2454-2470

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    Handle: RePEc:eee:csdana:v:56:y:2012:i:8:p:2454-2470
    Contact details of provider: Web page: http://www.elsevier.com/locate/csda

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    1. Christian Hennig, 2010. "Methods for merging Gaussian mixture 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. 4(1), pages 3-34, April.
    2. Wilfried Seidel & Hana Ševčíková, 2004. "Types of likelihood maxima in mixture models and their implication on the performance of tests," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 631-654, December.
    3. 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.
    4. Dankmar Böhning & Ekkehart Dietz & Rainer Schaub & Peter Schlattmann & Bruce Lindsay, 1994. "The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(2), pages 373-388, June.
    5. Yao, Weixin & Lindsay, Bruce G., 2009. "Bayesian Mixture Labeling by Highest Posterior Density," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 758-767.
    6. 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.
    7. Biernacki, Christophe & Chrétien, Stéphane, 2003. "Degeneracy in the maximum likelihood estimation of univariate Gaussian mixtures with EM," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 373-382, February.
    8. Chen, Jiahua & Tan, Xianming, 2009. "Inference for multivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1367-1383, August.
    9. Chris Fraley & Adrian E. Raftery, 2007. "Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 24(2), pages 155-181, September.
    10. Gabriela Ciuperca & Andrea Ridolfi & Jérome 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.
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