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Asymptotic theory of the likelihood ratio test for the identification of a mixture. (English) Zbl 1061.62028
Summary: The problems that arise when using the likelihood ratio test for the identification of a mixture distribution are well known: non-identifiability of the parameters and null hypotheses corresponding to a boundary point of the parameter space. In their approach to the problem of testing homogeneity against a mixture with two components, J. K. Ghosh and P. K. Sen [L. Le Cam and R. Olshen (eds.), Proc. Berkeley Conf. Honor J. Neyman and J. Kiefer, 789–806 (1985)] took into account these specific problems. Under general assumptions, they obtained the asymptotic distribution of the likelihood ratio test statistic. However, their result requires a separation condition which is not completely satisfactory. We show that it is possible to remove this condition with assumptions which involve the second derivatives of the density only.

62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
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[1] Bickel, P.; Chernoff, H., Asymptotic distribution of the likelihood ratio statistic in a prototypical non regular problem, (), 83-96
[2] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[3] Bock, H.H., Probability models and hypotheses testing in partitioning cluster analysis, (), 377-453 · Zbl 1031.62504
[4] Bock, H.H., Clustering methods: from classical models to new approaches, Statist. transition, 5, 725-758, (2002)
[5] Chen, H.; Chen, J., Large sample distribution of the likelihood ratio test for normal mixtures, Statist. probab. lett., 52, 125-133, (2001) · Zbl 0981.62015
[6] Dacunha-castelle, D.; Gassiat, E., Testing in locally conic models and application to mixture models, ESAIM probab. statist., 1, 285-317, (1997) · Zbl 1007.62507
[7] Garel, B., Likelihood ratio test for univariate Gaussian mixture, J. statist. plann. inference, 96, 325-350, (2001) · Zbl 0972.62011
[8] Garel, B.; Goussanou, F., Removing separation conditions in a 1 against 3-components Gaussian mixture problem, (), 61-73 · Zbl 1032.62013
[9] Ghosh, J.K., 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 Conferences in Honor of Jerzy Neyman and Jack Kiefer, Vol. II. Wadsworth, Monterey, pp. 789-806. · Zbl 1373.62075
[10] Hartigan, J.A., 1985. A failure of likelihood asymptotics for normal mixtures. In: Le Cam, L.M., Olshen, R.A. (Eds.), Proceedings of the Berkeley Conferences in Honor of Jerzy Neyman and Jack Kiefer, Vol. II. Wadsworth, Monterey, pp. 807-810. · Zbl 1373.62070
[11] Lemdani, M.; Pons, O., Likelihood ratio tests in contamination models, Bernoulli, 5, 705-719, (1999) · Zbl 0929.62015
[12] Liu, X.; Shao, Y., Asymptotics for likelihood ratio tests under loss of identifiability, Ann. statist., 31, 807-832, (2003) · Zbl 1032.62014
[13] Liu, X.; Shao, Y., Asymptotics for the likelihood ratio test in a two-component normal mixture model, J. statist. plann. inference, 123, 61-81, (2004) · Zbl 1050.62025
[14] McLachlan, G.J., On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture, Appl. statist., 36, 318-324, (1987)
[15] McLachlan, G.J.; Peel, D., Finite mixture models, wiley series in probability and statistics, (2000), Wiley New York
[16] Redner, R.A., Note on the consistency of the maximum likelihood estimate for non identifiable distributions, Ann. statist., 9, 225-228, (1981) · Zbl 0453.62021
[17] Robert, C., Titterington, M., 2001. Statistical mixtures and latent-structure modelling. International Centre for Mathematical Sciences. http://www.ceremade.dauphine.fr/ xian/Mixture01.html.
[18] Thode, H.C.; Finch, S.J.; Mendell, N.R., Simulated percentage points for the null distribution of the likelihood ratio test for a mixture of two normals, Biometrics, 44, 1195-1201, (1988) · Zbl 0715.62040
[19] Titterington, D.M., Some recent research in the analysis of mixture distributions, Statistics, 21, 619-641, (1990) · Zbl 0714.62023
[20] Van der Vaart, W.; Wellner, J.A., Weak convergence and empirical processes—with applications to statistics, Springer series in statistics, (1996), Springer Berlin · Zbl 0862.60002
[21] Wolfe, J.H., 1971. A Monte-Carlo study of the sampling distribution of the likelihood ratio for mixtures of multinormal distributions. Technical Bulletin STB, 72-2 U.S. Nav. Pers. and Train. Res. Lab., San Diego.
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