Abstract
In recent years the applications of multivariate Poisson models have increased, mainly because of the gradual increase in computer performance. The multivariate Poisson model used in practice is based on a common covariance term for all the pairs of variables. This is rather restrictive and does not allow for modelling the covariance structure of the data in a flexible way. In this paper we propose inference for a multivariate Poisson model with larger structure, i.e. different covariance for each pair of variables. Maximum likelihood estimation, as well as Bayesian estimation methods are proposed. Both are based on a data augmentation scheme that reflects the multivariate reduction derivation of the joint probability function. In order to enlarge the applicability of the model we allow for covariates in the specification of both the mean and the covariance parameters. Extension to models with complete structure with many multi-way covariance terms is discussed. The method is demonstrated by analyzing a real life data set.
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References
Bennett J.E., Racine-Poon A., and Wakefield J.C. 1996. MCMC for nonlinear hierarchical models. In: Gilks W.R., Richardson S., and Spiegelhalter D.J. (Eds.), Markov Chain Monte Carlo in Practice. Chapman and Hall, New York, pp. 339–357.
Berkhout P. and Plug E. 2004. A bivariate Poisson count data model using conditional probabilities. Statistica Neerlandica 58: 349–364.
Chib S. and Greenberg E. 1995. Understanding the Metropolis-Hastings algorithm. The American Statistician 49: 327–335.
Chib S. and Winkelmann R. 2001. Markov chain Monte Carlo analysis of correlated count data. Journal of Business and Economic Statistics 19: 428–435.
Dellaportas P. and Smith A.F.M. 1993. Bayesian inference for generalized linear and proportional hazard models via Gibbs sampling. Applied Statistics 41: 443–459.
Dey D.K., Ghosh S.K., and Mallick B.K. 2000. Generalized Linear Models: A Bayesian Perspective. Marcel-Dekker, Inc., New York.
Efron B. and Tibshirani R.J. 1993. An Introduction to the Bootstrap, Chapman and Hall, New York.
Gilks W.R., Best N.G., and Tan K.K.C. 1995. Adaptive rejection metropolis sampling within gibbs sampling. Applied Statistics 44: 155–73.
Gilks W.R., Richardson S., and Spiegelhalter D.J. 1996. Markov Chain Monte Carlo in Practice. Chapman and Hall, New York.
Ho L.L. and Singer J.M. 2001. Generalized least squares methods for bivariate Poisson regression. Communications in Statistics, Theory and Methods 30: 263–277.
Ibrahim J.G. and Laud P.W. 1991. On Bayesian analysis of generalized linear models using Jeffreys's prior. Journal of the American Statistical Association 86: 981–986.
Johnson N., Kotz S., and Balakrishnan N. 1997. Discrete Multivariate Distributions, New York, Wiley.
Kano K. and Kawamura K. 1991. On recurrence relations for the probability function of multivariate generalized Poisson distribution. Communications in Statistics-Theory and Methods 20: 165–178.
Kocherlakota, S. and Kocherlakota, K. 1992. Bivariate Discrete Distributions, Marcel Dekker, New York.
Kocherlakota S. and Kocherlakota K. 2001. Regression in the bivariate Poisson distribution. Communications in Statistics-Theory and Methods 30: 815–827.
Karlis D. 2003. An EM algorithm for multivariate Poisson distribution and related models. Journal of Applied Statistics 30: 63–77.
Karlis D. and Ntzoufras J. 2003. Analysis of sports data using bivariate Poisson models. The Statistician 52: 381–393 .
Lee L.F. 2001. On the range of correlation coefficients of bivariate ordered discrete random variables. Econometrics Journal 17: 247–256
Li C.S, Lu J.C., Park J., Kim K., and Peterson J. 1999. Multivariate zero-inflated Poisson models and their applications. Technometrics 41: 29–38.
Mahamunulu D.M. 1967. A note on regression in the multivariate Poisson distribution. Journal of the American Statistical Association 62: 251–258
McLachlan G. and Krishnan T. 1997. The EM Algorithm and Extensions, Wiley, New York.
Ntzoufras I., Dellaportas P., and Forster J.J. 2003. Bayesian variable and link determination for generalised linear models. Journal of Statistical Planning and Inference 111: 165–180.
Roberts G.O., Gelman A., and Gilks W. 1997. Weak convergence and optimal scaling of random walk Metropolis algorithms. Annals of Applied Probability 7: 110–120.
Roberts G.O. and Rosenthal S. 2001. Optimal scaling for various Metropolis-Hastings algorithms. Statistical Science 16: 351–367.
Tanner M.A. and Wong W.H. 1987. The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association 82: 528–540.
Tierney L. 1994. Markov chains for exploring posterior distributions (with discussion). The Annals of Statistics 22: 1701–1762.
Tsionas E.G. 1999. Bayesian analysis of the multivariate Poisson distribution. Communications in Statistics-Theory and Methods 28: 431–451.
Tsionas E.G. 2001. Bayesian multivariate Poisson regression. Communications in Statistics—Theory and Methods 30: 243–255.
Van Ophem H. 1999. A general method to estimate correlated discrete random variables. Econometric Theory 15: 228–237
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Karlis, D., Meligkotsidou, L. Multivariate Poisson regression with covariance structure. Stat Comput 15, 255–265 (2005). https://doi.org/10.1007/s11222-005-4069-4
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DOI: https://doi.org/10.1007/s11222-005-4069-4