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Root selection in normal mixture models

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  • Seo, Byungtae
  • Kim, Daeyoung

Abstract

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.

Suggested Citation

  • Seo, Byungtae & Kim, Daeyoung, 2012. "Root selection in normal mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2454-2470.
    • Handle: RePEc:eee:csdana:v:56:y:2012:i:8:p:2454-2470
    DOI: 10.1016/j.csda.2012.01.022
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    1. 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.
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    7. 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.
    8. 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.
    9. 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.
    10. 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.
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    Cited by:

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    2. 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.
    3. Heather Shappell & Sean L. Simpson, 2022. "Discussion on “Distributional independent component analysis for diverse neuroimaging modalities” by Ben Wu, Subhadip Pal, Jian Kang, and Ying Guo," Biometrics, The International Biometric Society, vol. 78(3), pages 1106-1108, September.
    4. Galimberti, Giuliano & Soffritti, Gabriele, 2014. "A multivariate linear regression analysis using finite mixtures of t distributions," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 138-150.
    5. 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.
    6. Xuwen Zhu & Volodymyr Melnykov, 2015. "Probabilistic assessment of model-based clustering," 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. 9(4), pages 395-422, December.
    7. Xiang, Sijia & Yao, Weixin & Seo, Byungtae, 2016. "Semiparametric mixture: Continuous scale mixture approach," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 413-425.
    8. 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.
    9. Krzysztof Podgórski & Jonas Wallin, 2015. "Maximizing leave-one-out likelihood for the location parameter of unbounded densities," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(1), pages 19-38, February.
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    11. Roberto Mari & Roberto Rocci & Stefano Antonio Gattone, 2020. "Scale-constrained approaches for maximum likelihood estimation and model selection of clusterwise linear regression models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 49-78, March.

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