×

On the consistency of MLE in finite mixture models of exponential families. (English) Zbl 1102.62020

Summary: Finite mixtures of densities from an exponential family are frequently used in the statistical analysis of data. Modelling by finite mixtures of densities from different exponential families provides more flexibility in the fittings, and get better results. However, in mixture problems, the log-likelihood function very often does not have an upper bound and therefore a global maximum does not always exist. R. A. Redner and H. F. Walker [Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev. 26, 195–239 (1984; Zbl 0536.62021)] provide conditions to assure the existence, consistency and asymptotic normality of the maximum likelihood estimator.These conditions are not generally easy to check, even for mixtures of densities from exponential families and, especially, from different exponential families. In this paper, results are given which make verification of the conditions easier in both cases.

MSC:

62F12 Asymptotic properties of parametric estimators
62E10 Characterization and structure theory of statistical distributions

Citations:

Zbl 0536.62021
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Al-Hussaini, E.K.; Abd-El-Hakim, N.S., Estimation of parameters of the inverse gaussian – weibull mixture model, Comm. statist. theory methods, 19, 5, 1607-1622, (1990)
[2] Andersen, T.G.; Bollerslev, T.; Diebold, F.X.; Labys, P., Modeling and forecasting realized volatility. econometrica, 71, 2, 579-625, (2003) · Zbl 1142.91712
[3] Atienza, N., 2005. Fitting the variable “Length of Hospital Stay” with mixtures from different distribution families (Available at ).
[4] Atienza, N., García-Heras, J., Muñoz-Pichardo, J.M., 2005a. A new condition for identifiability of finite mixture distributions, Metrika, to appear, doi:10.100.7/s00184-005-0013-z.
[5] Atienza, N., García-Heras, J., Muñoz-Pichardo, J.M., Villa, R., 2005b. Technical Report: Some Integrability Results on Exponential Families (Available at ).
[6] Barndorff-Nielsen, O., Information and exponential families in statistical theory, (1978), Wiley New York · Zbl 0387.62011
[7] Chanda, S., A note on the consistency and maxima of the roots of the likelihood equations, Biometrika, 41, 56-61, (1954) · Zbl 0055.12901
[8] Chandrasekhar, S.; Natarajan, R., Confidence limits for steady-state availability of systems with a mixture of exponential and gamma operating time and lognormal repair time, Microelectron. reliab., 36, 9, 1303-1304, (1996)
[9] Cheng, R.C.H.; Liu, W.B., The consistency of estimators in finite mixture models, Scand. J. statist., 28, 603-616, (2001) · Zbl 1010.62023
[10] Ciuperca, G.; Ridolfi, A.; Idier, J., Penalized maximum likelihood estimator for normal mixture, Scand. J. statist., 30, 45-59, (2003) · Zbl 1034.62018
[11] Cramér, S., Mathematical methods of statistics, (1946), Princeton University Press Princeton, NY · Zbl 0063.01014
[12] Everitt, B.S.; Hand, D.J., Finite mixture distributions, (1981), Chapman & Hall London · Zbl 0466.62018
[13] Feng, Z.D.; McCulloch, C.E., Statistical inference using maximum likelihood estimation and the generalized likelihood ratio when the true parameter is on the boundary of the parameter space, Statist. probab. lett., 13, 4, 325-332, (1992) · Zbl 0741.62034
[14] Feng, Z.D.; McCulloch, C.E., Using bootstrap likelihood ratios in finite mixture models, J. roy. statist. soc. ser. B, 58, 609-617, (1996) · Zbl 0906.62021
[15] Grodzenskii, S.Y.; Domrachev, V.G., Estimation of the parameters of a mixture of exponential and Weibull distributions with progressive censoring, Measurement tech., 45, f11, 1115-1118, (2002)
[16] Hathaway, R.J., A constrained formulation of maximum likelihood estimation for normal mixture distributions, Ann. statist., 13, 795-800, (1985) · Zbl 0576.62039
[17] Henna, J., Examples of identifiable mixture, J. Japan statist. soc., 24, 193-200, (1994) · Zbl 0818.62051
[18] Jewell, N.P., Mixtures of exponential distributions, Ann. statist., 10, 2, 479-484, (1982) · Zbl 0495.62042
[19] Leroux, S., Consistent estimation of a mixing distribution, Ann. statist., 20, 3, 1350-1360, (1992) · Zbl 0763.62015
[20] Lindsay, B.G., Mixture models: theory, geometry and applications, (1995), IMS · Zbl 1163.62326
[21] Marazzi, A.; Paccaud, F.; Ruffieux, C.; Beguin, C., Fitting the distributions of length of stay by parametric models, Medical care, 36, 6, 915-927, (1998)
[22] McLachlan, G.J., Krisnan, T., 1997. The EM Algorithm and Extensions. Wiley, New York.
[23] McLachlan, G.J.; Peel, D., Finite mixture models, (2001), Wiley New York
[24] Peters, B.C.; Walker, H.F., An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions, SIAM J. appl. math., 35, 362-378, (1978) · Zbl 0443.65112
[25] Pfanzagl, S., Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures, J. statist. plann. inference, 19, 137-158, (1988) · Zbl 0656.62044
[26] Rachev, S.T.; SenGupta, A., Laplace-Weibull mixtures for modeling price changes, Managemenent sci., 39, 8, 1029-1038, (1993) · Zbl 0785.90024
[27] Redner, R.A., Note on the consistency of the maximum-likelihood estimate for non-indentifiable distributions, Ann. statist., 9, 225-228, (1981) · Zbl 0453.62021
[28] Redner, R.A.; Walker, H.F., Mixture densities, maximum likelihood and the EM algorithm, SIAM rev., 26, 195-239, (1984) · Zbl 0536.62021
[29] Self, S.; Liang, K., Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions, J. amer. statist. assoc., 82, 398, 605-610, (1987) · Zbl 0639.62020
[30] Teicher, H., Identifiability of finite mixtures, Ann. math. statist., 34, 1265-1269, (1963) · Zbl 0137.12704
[31] Titterington, D.M.; Smith, A.F.M.; Makov, U.E., Statistical analysis of finite mixture distributions, (1990), Wiley New York · Zbl 0646.62013
[32] Van De Geer, S., Asymptotic theory for maximum likelihood in nonparametric mixture models, Comput. statist. data anal., 41, 453-464, (2003) · Zbl 1429.62117
[33] Wald, A., Note on the consistency of the maximum-likelihood estimate, Ann. math. statist., 20, 595-600, (1949) · Zbl 0034.22902
[34] Zasada, M.; Cieszewski, C.J., A finite mixture distribution approach for characterizing tree diameter distributions by natural social class in pure even-aged scots pine stands in Poland, Forest ecol. manage., 204, 2-3, 145-158, (2005)
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.