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Testing for a finite mixture model with two components

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  • Hanfeng Chen
  • Jiahua Chen
  • John D. Kalbfleisch

Abstract

Summary. We consider a finite mixture model with k components and a kernel distribution from a general one‐parameter family. The problem of testing the hypothesis k=2 versusk3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.

Suggested Citation

  • Hanfeng Chen & Jiahua Chen & John D. Kalbfleisch, 2004. "Testing for a finite mixture model with two components," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 95-115, February.
    • Handle: RePEc:bla:jorssb:v:66:y:2004:i:1:p:95-115
    DOI: 10.1111/j.1467-9868.2004.00434.x
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    1. Date, Paresh & Mamon, Rogemar & Tenyakov, Anton, 2013. "Filtering and forecasting commodity futures prices under an HMM framework," Energy Economics, Elsevier, vol. 40(C), pages 1001-1013.
    2. Zhu, Hongtu & Zhang, Heping, 2006. "Asymptotics for estimation and testing procedures under loss of identifiability," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 19-45, January.
    3. Vadim Marmer, 2008. "Testing the null hypothesis of no regime switching with an application to GDP growth rates," Empirical Economics, Springer, vol. 35(1), pages 101-122, August.
    4. Hiroyuki Kasahara & Tatsuyoshi Okimoto & Katsumi Shimotsu, 2014. "Modified Quasi-Likelihood Ratio Test for Regime Switching," The Japanese Economic Review, Japanese Economic Association, vol. 65(1), pages 25-41, March.
    5. Yang Ning & Yong Chen, 2015. "A Class of Pseudolikelihood Ratio Tests for Homogeneity in Exponential Tilt Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 504-517, June.
    6. Stehlík, Milan, 2008. "Homogeneity and scale testing of generalized gamma distribution," Reliability Engineering and System Safety, Elsevier, vol. 93(12), pages 1809-1813.
    7. Chen, Xiaohong & Ponomareva, Maria & Tamer, Elie, 2014. "Likelihood inference in some finite mixture models," Journal of Econometrics, Elsevier, vol. 182(1), pages 87-99.
    8. Andrew Sweeting, 2009. "The strategic timing incentives of commercial radio stations: An empirical analysis using multiple equilibria," RAND Journal of Economics, RAND Corporation, vol. 40(4), pages 710-742, December.
    9. Hiroyuki Kasahara & Katsumi Shimotsu, 2017. "Testing the Order of Multivariate Normal Mixture Models," CIRJE F-Series CIRJE-F-1044, CIRJE, Faculty of Economics, University of Tokyo.
    10. Sebastian Vollmer & Hajo Holzmann & Florian Schwaiger, 2013. "Peak vs Components," Review of Development Economics, Wiley Blackwell, vol. 17(2), pages 352-364, May.
    11. Hung-Chia Chen & James J. Chen, 2016. "Hybrid Mixture Model for Subpopulation Identification," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 8(1), pages 28-42, June.
    12. Meng Li & Sijia Xiang & Weixin Yao, 2016. "Robust estimation of the number of components for mixtures of linear regression models," Computational Statistics, Springer, vol. 31(4), pages 1539-1555, December.
    13. María Gallegos & Gunter Ritter, 2009. "Trimming algorithms for clustering contaminated grouped data and their robustness," 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. 3(2), pages 135-167, September.
    14. Kerby Shedden & Robert Zucker, 2008. "Regularized Finite Mixture Models for Probability Trajectories," Psychometrika, Springer;The Psychometric Society, vol. 73(4), pages 625-646, December.
    15. Lu, Zeng-Hua, 2009. "Covariate selection in mixture models with the censored response variable," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2710-2723, May.
    16. Da Silva Catela, Eva Yamila & Porcile, Gabriel & Gonçalves, Flávio, 2010. "Brazilian municipalities: agglomeration economies and development levels in 1997 and 2007," Revista CEPAL, Naciones Unidas Comisión Económica para América Latina y el Caribe (CEPAL), August.
    17. Pittau, Maria Grazia & Zelli, Roberto & Johnson, Paul, "undated". "Mixture Models and Convergence Clubs," Vassar College Department of Economics Working Paper Series 91, Vassar College Department of Economics.
    18. 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.
    19. Christian Ritz, 2013. "Penalized likelihood ratio tests for repeated measurement models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 534-547, September.
    20. Maria Grazia Pittau & Roberto Zelli & Paul A. Johnson, 2010. "Mixture Models, Convergence Clubs, And Polarization," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 56(1), pages 102-122, March.
    21. Cui, Yin & Fu, Yuejiao & Hussein, Abdulkadir, 2009. "Group sequential testing of homogeneity in genetic linkage analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(10), pages 3630-3639, August.
    22. Hajo Holzmann & Sebastian Vollmer & Julian Weisbrod, 2007. "Twin Peaks or Three Components? - Analyzing the World\'s Cross-Country Distribution of Income," Ibero America Institute for Econ. Research (IAI) Discussion Papers 162, Ibero-America Institute for Economic Research.
    23. Gallegos, María Teresa & Ritter, Gunter, 2010. "Using combinatorial optimization in model-based trimmed clustering with cardinality constraints," Computational Statistics & Data Analysis, Elsevier, vol. 54(3), pages 637-654, March.
    24. Dannemann, Jörn & Holzmann, Hajo, 2010. "Testing for two components in a switching regression model," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1592-1604, June.

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