:: Volume 15, Issue 2 (3-2019) ::
JSRI 2019, 15(2): 173-212 Back to browse issues page
Model Selection for Mixture Models Using Perfect Sample
Sadegh Fallahigilan1, Abdolreza Sayyareh *2
1- Razi University
2- K. N. Toosi University of Technology , asayyareh@kntu.ac.ir
Abstract:   (924 Views)
We have considered a perfect sample method for model selection of finite mixture models with either known (fixed) or unknown number of components which can be applied in the most general setting with assumptions on the relation between the rival models and the true distribution. It is, both, one or neither to be well-specified or mis-specified, they may be nested or non-nested. We consider mixture distribution as a complete-data (bivariate) distribution by prediction of missing data variable (unobserved variable) and show that this ideas is applicable to use Vuong's test for select optimum mixture model when number of components are known (fixed) or unknown. We have considered  AIC  and  BIC  based on the complete-data distribution. The performance of this method is evaluated by Monte-Carlo method and real data set, as Total Energy Production. 
Keywords: finite mixture model, perfect sample, model selection, missing data variable, Vuong's test.
Full-Text [PDF 1869 kb]   (530 Downloads)    
Type of Study: Research | Subject: General
Received: 2018/06/17 | Accepted: 2019/07/20 | Published: 2019/03/28
1. Akaike, H. (1973). Information Theory and an Extension of Maximum Likelihood Principle. Second International Symposium on Information Theory, Akademia Kiado, 267-281.
2. Chen, H., Chen, J. and Kalbfeisch, J.D. (2001). A Modified Likelihood Ratio Test for Homogeneity in the Finite Mixture Models. Journal of the Royal Statistical Society. Series B (Statistical Methodology). 63, 19-29. [DOI:10.1111/1467-9868.00273]
3. Chen, J. and Kalbfleisch, J.D. (2005). Modified Likelihood Ratio Test in Finite Mixture Models with a Structural Parameter. Journal of statistical Planning and Inference, 129, 93-107. [DOI:10.1016/j.jspi.2004.06.041]
4. Chernoff, H. and Lander, E. (1995). Asymptotic Distribution of the Likelihood Ratio Test that a Mixture of Two Binomials is a Single Binomial. Journal of Statistical Planning and Inference, 43, 19-40. [DOI:10.1016/0378-3758(94)00006-H]
5. Crawford, S.L. (1994). An Application of the Laplace Method to Finite Mixture Distributions. Journal of the American Statistical Association, 89, 259-267. [DOI:10.1080/01621459.1994.10476467]
6. Dacunha-Castelle, D. and Gassiat, E. (1999). Testing the Order of a Model using Locally Conic Parametrization: Population Mixtures and Stationary ARMA Processes. The Annals of Statistics, 27, 1178-1209. [DOI:10.1214/aos/1017938921]
7. Everitt, B.S. and David, J. Hand (1981). Finite Mixture Distributions. Monographs on Applied Probability and Statistics. Chapman and Hall, London, New York. [DOI:10.1007/978-94-009-5897-5]
8. Fallahigilan, S. and Sayyareh, A. (2016). Finite Mixture Model Selection for Total Energy Consumption. International Journal of Energy and Statistics, 4. [DOI:10.1142/S2335680416500095]
9. Feng, Z.D. and McCulloch, C.E. (1994). On the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture with Unequal Variances. Biometrics, 1158-1162. [DOI:10.2307/2533453]
10. Ghosh, J.H. and Sen, P.K. (1985). On the Asymptotic Performance of the Log Likelihood Ratio Statistic for the Mixture Model and Related Results. In: Le Cam, L.M., Olshen, R.A. (Eds.) Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. II. Wadsworth, Monterey, pp. 789-806.
11. Jeffries, N.O. (2003). A Note on Testing the Number of Components in a Normal Mixture. Biometrika, 90, 991-994. [DOI:10.1093/biomet/90.4.991]
12. Kazakos, D. (1977). Recursive Estimation of Prior Probabilities using a Mixture. IEEE Transactions on Information Theory, 23, 203-211. [DOI:10.1109/TIT.1977.1055693]
13. Lo, Y. (2005). Likelihood Ratio Tests of the Number of Components in a Normal Mixture with Unequal Variances. Statistics & probability letters, 71, 225-235. [DOI:10.1016/j.spl.2004.11.007]
14. Lo, Y., Mendell, N.R. and Rubin, D.B. (2001). Testing the Number of Components in a Normal Mixture. Biometrika, 88, 767-778. [DOI:10.1093/biomet/88.3.767]
15. McLachlan, G.J. and Basford, K.E. (1988). Mixture Models: Inference and Applications to Clustering, 84. [DOI:10.2307/2289892]
16. McLachlan, G. and Peel, D. (2004). Finite Mixture Models. John Wiley & Sons.
17. Morgan, G.B. (2015). Mixed Mode Latent Class Analysis: An Examination of Fit Index Performance for Classification. Structural Equation Modeling: A Multidisciplinary Journal, 22, 76-86. [DOI:10.1080/10705511.2014.935751]
18. Morgan, G.B., Hodge, K.J. and Baggett, A.R. (2016). Latent Profile Analysis with Nonnormal Mixtures: A Monte Carlo Examination of Model Selection using Fit Indices. Computational Statistics & Data Analysis, 93, 146-161. [DOI:10.1016/j.csda.2015.02.019]
19. Nylund, K.L., Asparouhov, T. and Muthen, B.O. (2008). Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study: Erratum. [DOI:10.1080/10705510701575396]
20. Redner, R.A. and Walker, H.F. (1984). Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM review, 26, 195-239. [DOI:10.1137/1026034]
21. Sayyareh, A. (2016). Admissible Set of Rival Models based on the Mixture of Kullback-Leibler Risks. JSRI. 13, 59-88 [DOI:10.18869/acadpub.jsri.13.1.4]
22. Schwarz, G. (1978). Estimating the Dimension of a Model. The Annals of Statistics, 6, 461-464. [DOI:10.1214/aos/1176344136]
23. Scott, A.J. and Symons, M.J. (1971). Clustering Methods based on Likelihood Ratio Criteria. Biometrics, 387-397. [DOI:10.2307/2529003]
24. Symons, M.J. (1981). Clustering Criteria and Multivariate Normal Mixtures. Biometrics, 35-43. [DOI:10.2307/2530520]
25. Teicher, H. (1961). Identifiability of Mixtures. The Annals of Mathematical Statistics, 32, 244-248. [DOI:10.1214/aoms/1177705155]
26. Titterington, D.M., Smith, A.F.M. and Makov, U.E. (1985). Statistical Analysis of Finite Mixture Distributions. Wiley, New York.
27. Vuong, Q.H. (1989). Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses. Econometrica: Journal of the Econometric Society, 307-333. [DOI:10.2307/1912557]
28. Wald, A. (1948). Estimation of a Parameter when the Number of Unknown Parameters Increases Indefinitely with the Number of Observations. The Annals of Mathematical Statistics, 220-227. [DOI:10.1214/aoms/1177730246]
29. White, H. (1982). Maximum Likelihood Estimation of Mis-specified Models. Econometrica: Journal of the Econometric Society, 1-25. [DOI:10.2307/1912526]
30. Wichitchan, S., Yao, W. and Yang, G. (2018). Hypothesis Testing for Finite Mixture Models. Computational Statistics & Data Analysis. [DOI:10.1016/j.csda.2018.05.005]
31. Yakowitz, S.J. and Spragins, J.D. (1968). On the Identifiability of Finite Mixtures. The Annals of Mathematical Statistics, 209-214. [DOI:10.1214/aoms/1177698520]

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