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A Multivariate Discrete Poisson-Lindley Distribution: Extensions and Actuarial Applications

Published online by Cambridge University Press:  09 August 2013

José María Sarabia
Affiliation:
Department of Economics, University of Cantabria, Santander, Spain
N. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, CanadaL8S 4K1

Abstract

This paper proposes multivariate versions of the continuous Lindley mixture of Poisson distributions considered by Sankaran (1970). This new class of distributions can be used for modelling multivariate dependent count data when marginal overdispersion is present. After discussing some of its properties, a general multivariate model with Poisson-Lindley marginals and with a flexible covariance structure is proposed. Several specific models as well as one that allows correlations of any sign are considered, and then some estimation methods are discussed. Finally, some illustrative examples are given for fitting and demonstrating the usefulness of these bivariate distributions.

Type
Research Article
Copyright
Copyright © International Actuarial Association 2012

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References

Aitchison, J. and Ho, C.H. (1989) The multivariate Poisson-Log normal distribution. Biometrika, 76, 643653.CrossRefGoogle Scholar
Arbous, A.G. and Sichel, H.S. (1954) New techniques for the analysis of absenteeism data. Biometrika, 41, 7790.CrossRefGoogle Scholar
Barndorff-Nielsen, P., Blaesild, P. and Seshadri, V. (1992) Multivariate distributions with generalized inverse Gaussian marginals and associated Poisson mixtures. The Canadian Journal of Statistics, 20, 109120.CrossRefGoogle Scholar
Bermúdez, L. (2009) A priori ratemaking using bivariate Poisson regression models. Insurance: Mathematics and Economics, 44, 135141.Google Scholar
Boucher, J.P., Denuit, M. and Guillén, M. (2009) Number of accidents or number of claims? An approach with zero-inflated Poisson models for panel data. The Journal of Risk and Insurance, 76(4), 821846.CrossRefGoogle Scholar
Farlie, D.J. (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47, 307323.CrossRefGoogle Scholar
Frees, E.W. (2004) Longitudinal and Panel Data. Analysis and Applications in the Social Sciences. Cambridge University Press.CrossRefGoogle Scholar
Ghitany, M.E. and Al-Mutairi, L. (2009) Estimation methods for the discrete Poisson-Lindley distribution. Journal of Statistical Computation and Simulation, 79, 19.CrossRefGoogle Scholar
Ghitany, M.E., Atieh, B. and Nadarajah, S. (2008) Lindley distribution and its application. Mathematics and Computers in Simulation, 78, 493506.CrossRefGoogle Scholar
Gómez-Déniz, E., Sarabia, J.M. and Calderín, E. (2008) Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications. Insurance: Mathematics and Economics, 42, 3949.Google Scholar
Hesselager, O. (1996) Recursions for certain bivariate counting distributions and their compound distributions. Astin Bulletin, 26, 3552.CrossRefGoogle Scholar
Holla, M.S. and Bhattacharya, S.K. (1965) On a discrete compound Poisson distribution. Annals of the Institute of Statistical Mathematics, 15, 377384.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions, John Wiley and Sons, New York.Google Scholar
Karlis, D. and Meligkotsidou, L. (2007) Finite mixtures of multivariate Poisson distributions with application. Journal of Statistical Planning and Inference, 137, 19421960.CrossRefGoogle Scholar
Karlis, D. and Xekalaki, E. (2005) Mixed Poisson distributions. International Statistical Review, 73, 3558.CrossRefGoogle Scholar
Kemp, A.W. (1981) Computer sampling from homogeneous bivariate discrete distributions. ASA Proceedings of the Statistical Computing Section, 173175.Google Scholar
Kocherlakota, S. and Kocherlakota, K. (1992) Bivariate Discrete Distributions, Marcel Dekker, New York.Google Scholar
Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions, Vol. 1: Models and Applications. John Wiley and Sons, New York.CrossRefGoogle Scholar
Lai, C.D. (2006) Constructions of discrete bivariate distributions. In: Advances on Distribution Theory, Order Statistics and Inference (Eds., Balakrishnan, N., Castillo, E., Sarabia, J.M.), pp. 2958, Birkhäuser, Boston.CrossRefGoogle Scholar
Lee, M.-L.T. (1996) Properties and applications of the Sarmanov family of bivariate distributions. Communications in Statistics-Theory and Methods, 25, 12071222.Google Scholar
Lindley, D.V. (1958) Fiducial distributions and Bayes's theorem. Journal of the Royal Statistical Society, Series B, 20, 102107.Google Scholar
Øigard, T.A. and Hanssen, A. (2002) The multivariate normal inverse Gaussian heavy-tailed distribution: simulation and estimation. In: IEEE International Conference on Acoustics, Speech and Signal Processing, 2, 14891492.Google Scholar
Partrat, C. (1994) Compound model for two dependent kinds of claim. Insurance: Mathematics and Economics, 15, 219231.Google Scholar
Protassov, R.S. (2004) EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ . Statistics and Computing, 14, 6777.CrossRefGoogle Scholar
Sarabia, J.M. and Castillo, E. (2006) Bivariate distributions based on the generalized three-parameter beta distribution. In: Advances on Distribution Theory, Order Statistics and Inference, (Eds., Balakrishnan, N., Castillo, E., Sarabia, J.M.), pp. 85110, Birkhäuser, Boston.CrossRefGoogle Scholar
Sarabia, J.M. and Gómez-Déniz, E. (2011) Construction of multivariate distributions: a review of some recent results. Statistics and Operational Research Transactions (SORT), 32, 336 (with discussion).Google Scholar
Sarabia, J.M. and Gómez-Déniz, E. (2011) Multivariate Poisson-Beta distributions with applications. Communications in Statistics-Theory and Methods, 40, 10931108.CrossRefGoogle Scholar
Sankaran, M. (1970) The discrete Poisson-Lindley distribution. Biometrics, 26, 145149.CrossRefGoogle Scholar
Sarmanov, O.V. (1966) Generalized normal correlation and two-dimensional Frechet classes. Doklady, Soviet Mathematics, 168, 596599.Google Scholar
Sichel, H.S. (1971) On a family of discrete distributions particularly suited to represent long-tailed frequency data. In: Proceedings of the Third Symposium on Mathematical Statistics (Eds. N.F. Laubscher), Council for Scientific and Industrial Research, pp. 5197, Pretoria, South Africa.Google Scholar
Stein, G., Zucchini, W. and Juritz, J. (1987) Parameter estimation for the Sichel distribution and its multivariate extension. Journal of the American Statistical Association, 82, 938944.CrossRefGoogle Scholar
Sundt, B. (2002) Recursive evaluation of aggregate claims distributions. Insurance: Mathematics and Economics, 30, 297322.Google Scholar
Thyrion, P. (1972) Quelques observations statistiques sur la variable, nombre de sinistres en assurance automobile. Astin Bulletin, VI(3), 203232.CrossRefGoogle Scholar
Vernic, R. (1997) On the bivariate generalized Poisson distribution. Astin Bulletin, 27, 2331.CrossRefGoogle Scholar
Vernic, R. (1999) Recursive evaluation of some bivariate compound distributions. Astin Bulletin, 29, 315325.CrossRefGoogle Scholar
Walhin, J.F. and Paris, J. (2000a) A general family of bivariate mixed Poisson distributions. www.stat.ucl.ac.be/ISpub/dp/2000/dp0011.ps Google Scholar
Walhin, J.F. and Paris, J. (2000b) Recursive formulae for some bivariate counting distributions obtained by the trivariate reduction method. Astin Bulletin, 30, 141155.CrossRefGoogle Scholar
Walhin, J.F. and Paris, J. (2001) The mixed bivariate Hoffmann distribution. Astin Bulletin, 31, 123138.CrossRefGoogle Scholar

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