A stochastic EM algorithm for a semiparametric mixture model. (English) Zbl 1445.62056

Summary: Recently, there has been a considerable interest in finite mixture models with semi-/nonparametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. Hence, a generalization of the EM algorithm to semiparametric mixture models is proposed. The approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the proposed EM type estimators is studied numerically not only through several Monte Carlo experiments but also through comparison with alternative methods existing in the literature. In addition to these numerical experiments, applications to real data are provided, showing that the estimation method behaves well, that it is fast and easy to be implemented.


62G05 Nonparametric estimation
62F12 Asymptotic properties of parametric estimators
Full Text: DOI Link


[1] Bordes, L.; Mottelet, S.; Vandekerkhove, P., Semiparametric estimation of a two-component mixture model, Ann. statist., 34, 1204-1232, (2005) · Zbl 1112.62029
[2] Bordes, L., Delmas, C., Vandekerkhove, P., 2006. Semiparametric estimation of a two-component mixture model where a component is known. Scand. J. Statist., to appear. · Zbl 1164.62331
[3] Celeux, G.; Diebolt, J., A stochastic approximation type EM algorithm for the mixture problem, Stoch. stoch. rep., 41, 119-134, (1992) · Zbl 0766.62050
[4] Celeux, G.; Chauveau, D.; Diebolt, J., Stochastic versions of the EM algorithm: an experimental study in the mixture case, J. statist. comput. simul., 55, 287-314, (1996) · Zbl 0907.62024
[5] Chauveau, D., A stochastic EM algorithm for mixtures with censored data, J. statist. plann. inference, 46, 1-25, (1995) · Zbl 0821.62013
[6] Cruz-Medina, I.R.; Hettmansperger, T.P., Nonparametric estimation in semi-parametric univariate mixture models, J. statist. comput. simul., 74, 513-524, (2004) · Zbl 1060.62041
[7] Dacunha-Castelle, D.; Gassiat, E., Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes, Ann. statist., 27, 1178-1209, (1999) · Zbl 0957.62073
[8] Dempster, A.; Laird, N.; Rubin, D., Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. roy. statist. soc. ser. B, 39, 1-38, (1977) · Zbl 0364.62022
[9] Hall, P., On the nonparametric estimation of mixture proportions, J. roy. statist. soc. ser. B, 43, 147-156, (1981) · Zbl 0472.62052
[10] Hall, P.; Zhou, X.-H., Nonparametric estimation of component distributions in a multivariate mixture, Ann. statist., 31, 201-224, (2003) · Zbl 1018.62021
[11] Hettmansperger, T.P.; Thomas, H., Almost nonparametric inference for repeated measures in mixture models, J. roy. statist. soc. ser. B, 62, 811-825, (2000) · Zbl 0957.62026
[12] Hunter, D.R., Wang, S., Hettmansperger, T.P., 2004. Inference for mixtures of symmetric distributions. Ann. Statist., to appear. · Zbl 1114.62035
[13] Lemdani, M.; Pons, O., Likelihood ratio tests in contamination models, Bernoulli, 5, 705-719, (1999) · Zbl 0929.62015
[14] Leroux, B.G., Consistent estimation of a mixing distribution, Ann. statist., 20, 1350-1360, (1992) · Zbl 0763.62015
[15] Lindsay, B.G., 1995. Mixture Models: Theory, Geometry and Applications. NSFCBMS Regional Conference Series in Probability and Statistics, vol. 5, IMS, ASA. · Zbl 1163.62326
[16] McLachlan, G.; Peel, D.A., Finite mixture models, (2000), Wiley New York · Zbl 0963.62061
[17] McLachlan, G.J.; Krishnan, T., The EM algorithm and extensions, (1997), Wiley New York · Zbl 0882.62012
[18] McNeil, D.R., Interactive data analysis, (1977), Wiley New York
[19] Redner, R.A.; Walker, H.F., Mixtures densities, maximum likelihood and the EM algorithm, SIAM rev., 26, 195-249, (1984) · Zbl 0536.62021
[20] Robin, S., Bar-Hen, A., Daudin, J.J., 2005. A semiparametric approach for mixture models: Application to local FDR estimation. preprint INA/INRIA, France. · Zbl 1445.62075
[21] Titterington, D.M., Minimum-distance non-parametric estimation of mixture proportions, J. roy. statist. soc. ser. B, 45, 37-46, (1983) · Zbl 0563.62027
[22] Titterington, D.M.; Smith, A.F.M.; Makov, U.E., Statistical analysis of finite mixture distributions, (1985), Wiley Chichester · Zbl 0646.62013
[23] Wei, G.C.G.; Tanner, M.A., A Monte Carlo implementation of the EM algorithm and the poor Man’s data augmentation algorithms, J. amer. statist. assoc., 85, 699-704, (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.