×

zbMATH — the first resource for mathematics

A stochastic EM algorithm for mixtures with censored data. (English) Zbl 0821.62013
Summary: The stochastic EM algorithm is a widely applicable approach for computing maximum likelihood estimates for the mixture problem. We present here an extension of the SEM algorithm in a particular case of incomplete data, where the loss of information is due both to mixture models and censored observations. We propose several solutions to implement the ‘SEMcm algorithm’ (SEM for censored mixture), showing in particular that one of these procedures solves numerical problems arising with the EMcm algorithm and mixtures of nonexponential-type distributions.
Theoretically, we study the asymptotic behavior of SEMcm in the simple case of a two-component censored mixture, where the unknown parameter is the mixing proportion. We prove, for each SEMcm procedure, convergence of the stationary distribution to a Gaussian distribution located on the m.l.e. of the parameter. To conclude, we give some examples based on simulations for censored samples with a great amount of lost information.

MSC:
62F12 Asymptotic properties of parametric estimators
65C99 Probabilistic methods, stochastic differential equations
62F10 Point estimation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[2] Celeux, G.; Diebolt, J., The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem, Comput. statist. quat., 2, 1, 73-82, (1985)
[3] Celeux, G.; Diebolt, J., A random imputation principle: the stochastic EM algorithm, INRIA, rapport de recherche, no. 901, (1988)
[4] Celeux, G.; Diebolt, J., Asymptotic properties of a stochastic EM algorithm for estimating mixing propositions, Commun. statist. stochastic models, 9, 599-613, (1993) · Zbl 0783.62021
[5] Chauveau, D., Extension des algorithmes EM et SEM à la reconnaissance de Mélanges censurés de distributions de Défaillances, ()
[6] Chauveau, D., Algorithmes EM et SEM pour un Mélange censuré de distributions de défailances, application à la fiabilité, Rev. statist. appl., 40, 67-76, (1992)
[7] Dempster, A.P.; Laird, N.M.; Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm, J.R. statist. soc. ser. B, 39, 1-38, (1977) · Zbl 0364.62022
[8] Duflo, M., Méthodes Récursives aléatoires, (1991), Masson · Zbl 0703.62084
[9] Redner, R.A.; Walker, H.F., Mixture densities, maximum likelihood and the EM algorithm, SIAM rev., 26, 2, 195-239, (1984) · Zbl 0536.62021
[10] Silverman, B.W., Some asymptotic properties of the probabilistic teacher, IEEE, inform, theory, 26, 2, 246-249, (1980) · Zbl 0428.62026
[11] Wu, C.F., On the convergence properties of the EM algorithm, Ann. statist., 11, 95-103, (1983) · Zbl 0517.62035
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.