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EM Algorithm for Mixed Poisson and Other Discrete Distributions

Published online by Cambridge University Press:  17 April 2015

Dimitris Karlis
Dimitris Karlis*
Affiliation:
Department of Statistics, Athens University of Economics and Business, 76, Patission str, 10434, Athens, Greece. Email: [email protected]
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Abstract

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Mixed Poisson distributions are widely used in various disciplines including actuarial applications. The family of mixed Poisson distributions contains several members according to the choice of the mixing distribution for the parameter of the Poisson distribution. Very few of them have been studied in depth, mainly because of algebraic intractability. In this paper we will describe an EM type algorithm for maximum likelihood estimation for mixed Poisson distributions. The main achievement is that it reduces the problem of estimation to one of estimation of the mixing distribution which is usually easier. Variants of the algorithm work even when the probability function of the mixed distribution is not known explicitly but we have only an approximation of it. Other discrete distributions are treated as well.

Keywords

Poisson mixturesbeta-binomial distributionYule distributionMonte Carlo EM
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 3 - 24
Copyright
Copyright © ASTIN Bulletin 2005

References

Aitkin, M. (1996) A General Maximum Likelihood Analysis of Overdispersion in Generalized Linear Models. Statistics and Computing, 6, 251–262.CrossRefGoogle Scholar
Booth, J.G. and Caffo, B.S. (2002) Unequal sampling for Monte Carlo EM algorithms. Computational Statistics and Data Analaysis, 39, 261–270.CrossRefGoogle Scholar
Booth, J.G. and Hobert, J.P. (1999) Maximizing generalized linear mixed model likelihood with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society, Series B, 61, 265–285.CrossRefGoogle Scholar
Bohning, D. (1999) Computer assisted analysis of mixtures and applications in meta-analysis, disease mapping and others. CRC Press, New York.Google Scholar
Bulmer, M. (1974) On fitting the Poisson-lognormal distribution to species-abundance data. Bio-metrics, 30, 101–110.Google Scholar
Carlin, B.P., and Louis, T.A. (1996) Bayes and Empirical Bayes Methods for Data Analysis. Great Britain: Chapman and Hall.Google Scholar
Celeux, G. and Diebolt, J. (1985) The SEM Algorithm, a Probabilistic Teacher Algorithm Derived from the EM Algorithm for the Mixture Problem. Computational Statistics Quarterly, 2, 73–82.Google Scholar
Chan, K.S. and Ledolter, J. (1995) Monte Carlo EM estimation for time series models involving counts. Journal of the American Statistical Association, 90, 242–252.CrossRefGoogle Scholar
Cowles, M.K. and Carlin, B.P. (1996) Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review. Journal of the American Statistical Association, 91, 883–904.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M. and Rubin, D. (1977) Maximum Likelihood from Incomplete Data Via the EM Algorithm. Journal of the Royal Statistical Society, B 39, 1–38.Google Scholar
Douglas, J.B. (1980) Analysis With Standard Contagious Distributions. Statistical distributions in scientific work series, Vol 4, ICPH, Maryland, USA.Google Scholar
Douglas, J.B. (1994) Empirical Fitting of Discrete Distributions. Biometrics, 50, 576–579.CrossRefGoogle Scholar
Goutis, C. (1993) Recovering Extra Binomial Variation. Journal of Statistical Computation and Simulation, 45, 233–242.CrossRefGoogle Scholar
Grandell, J. (1997) Mixed Poisson Processes. Chapman and Hall/CRC Statistics and Mathematics, New York.CrossRefGoogle Scholar
Griffiths, D.A. (1973) Maximum Likelihood Estimation for the Beta Binomial Distributions and an Application to the Household Distribution of the Total Number of Cases of a Disease. Biometrics, 29, 637–648.CrossRefGoogle Scholar
Greenwood, M. and Yule, G. (1920) An Inquiry Into the Nature of Frequency Distributions Representative of Multiple Happenings with Particular Reference to the Occurence of Multiple Attacks of Disease Or of Repeated Accidents. Journal of the Royal Statistical Society, A 83, 255–279.CrossRefGoogle Scholar
Johnson, N., Kotz, S. and Kemp, A.W. (1992) Univariate discrete distributions, Willey-New York.Google Scholar
Jorgensen, B. (1982) The generalized Inverse Gaussian distribution. Lecture Notes in Statistics, 9, Springer-Verlag.Google Scholar
Karlis, D. (1998) Estimation and hypothesis testing problems in Poisson mixtures. Phd Thesis, Department of Statistics, Athens University of Economics.Google Scholar
Karlis, D. (2001) Exact ML estimation for Mixed Poisson Regression Models. Statistical Modelling: An International Journal, 1, 305–319.CrossRefGoogle Scholar
Kemp, C.D. and Kemp, A.W. (1965) Some Properties of the Hermite Distribution. Biometrika, 52, 381–394.Google ScholarPubMed
Kemp, A.W. and Kemp, C.D. (1966) An Alternative Derivation of the Hermite Distribution, Biometrika, 53, 627–628.CrossRefGoogle Scholar
Levine, R.A. and Casella, G. (2001) Implementations of the Monte Carlo EM Algorithm. Journal of Computational and Graphical Statistics, 10, 422–439.CrossRefGoogle Scholar
Lindsay, B. (1995) Mixture Models: Theory, Geometry and Applications. Regional Conference Series in Probability and Statistics, Vol 5, Institute of Mathematical Statistics and American Statistical Association.CrossRefGoogle Scholar
McCulloch, P. (1997) Maximum Likelihood algorithms for generalized linear mixed models. Journal of the American statistical Association, 92, 162–170.CrossRefGoogle Scholar
McLachlan, G. and Peel, D. (2000) Finite mixture models, Wiley, New York.CrossRefGoogle Scholar
Meng, X.L. and Rubin, D. (1993) Maximum Likelihood Estimation via the ECM Algorithm: A General Framework. Biometrika, 80, 267–278.CrossRefGoogle Scholar
Munkin, M.K. and Trivedi, P.K. (1999) Simulated Maximum Likelihood Estimation of Multivariate Mixed-Poisson Regression Models, with Application. Econometrics Journal, 2, 29–48.CrossRefGoogle Scholar
Ong, S.H. (1995) Computation of Probabilities of a Generalised log-series and Related Distributions. Communication in Statistics – Theory and Methods, 24, 253–271.CrossRefGoogle Scholar
Panjer, H. (1981) Recursive Evaluation of a Family of Compound Distributions. ASTIN Bulletin, 18, 57–68.Google Scholar
Piegorsch, W.W. (1990) Maximum Likelihood Estimation for the Negative Binomial Dispersion Paramater. Biometrics, 46, 863–867.CrossRefGoogle Scholar
Sankaran, M. (1970) The Discrete Poisson-Lindley Distribution. Biometrics, 26, 145–149.CrossRefGoogle Scholar
Sapatinas, T. (1995) Identifiability of mixtures of power-series distributions and related characterizations, Annals of the Institute of Statistical Mathematics, 47, 447–459.CrossRefGoogle Scholar
Sichel, H.S. (1974) On a distribution representing sentence-length in written prose. Journal of the Royal Statistical Society, A 137, 25–34.CrossRefGoogle Scholar
Sichel, H.S. (1982) Asymptotic efficiencies of three methods of estimation for the inverse gaussian-Poisson distribution. Biometrika, 69, 467–472.CrossRefGoogle Scholar
Shaban, S.A. (1988) Poisson-Lognormal Distributions. In Crow, E.L. and Shimizu, K. (eds), Log-normal Distributions: Theory and Applications, 195–210, New York, Marcel and Dekker.Google Scholar
Sprott, D. (1983) Estimating the parameters of a convolution by the Maximum Likelihood. Journal of the American Statistical Association, 78, 457–460.CrossRefGoogle Scholar
Stewart, J.A. (1994) The Poisson-Lognormal Model for Bibliometric/Scientometric Distributions. Information Processing and Management, 30, 239–251.CrossRefGoogle Scholar
Tanner, M.A. (1996) Tools for statistical inference: Methods for the Exploration of Posterior distributions and Likelihood Functions, 3rd edition, Springer, New York.CrossRefGoogle Scholar
Titterington, D.M, Smith, A.F.M. and Makov, U.E. (1985) Statistical analysis of finite mixtures distributions. Willey and sons, New York.Google Scholar
Tripathi, R., Gupta, R. and Gurland, J. (1994) Estimation of the Parameters in the Beta Binomial Model. Annals of the Institute of Statistical Mathematics, 46, 317–331.CrossRefGoogle Scholar
Wang, Y. (1996) Estimation Problems for the Two-Parameter Negative Binomial Distribution. Statistics and Probability Letters, 26, 113–114.CrossRefGoogle Scholar
Wei, G.C.G. and Tanner, M.A. (1990) A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. Journal of the American Statistical Association, 85, 699–704.CrossRefGoogle Scholar
Willmot, G. (1993) On Recursive Evaluation of Mixed Poisson Probabilities and Related Quantities. Scandinavian Actuarial Journal, 101–113.Google Scholar