Heterogeneity in models of purchase frequency. A comparison of Poisson-gamma mixtures with finite Poisson mixtures
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Poisson models are fundamental in the modelling of purchase frequencies. However, very often they are statistically incompatible with the data. This stems from the fact that the mean is assumed to be equal to the variance and, in consequence, this fails to capture heterogeneity. Thus Poisson mixture models are often considered instead. The most commonly used of these models is the Poisson-gamma mixture model, which is very often applied to problems in marketing. Hence, it would be advisable to discover its limitations. Using real marketing data sets, we point out the limitations of this approach. Furthermore, we compare it with finite Poisson mixtures.
- BROCKETT P.L., GOLDEN L.L., PANJER H.H., Flexible purchase frequency modeling, Journal of Marketing Research, 1996, 33, 94–107.
- CAMERON A.C., TRIVEDI P.K., Regression analysis of count data. Cambridge University Press, 1998.
- DEMPSTER A.P., LAIRD N.M., RUBIN D.B., Maximum likelihood from incomplete data via EM algorithm, Journal of the Royal Statistical Society Series B-Methodological, 1977, 39, 1–38.
- DILLON W., KUMAR A. (1994), Advanced methods of marketing research, [In:] R.P. Bagozzi Ed.), Latent structure and other mixture models in marketing: An integrative survey and overview, Oxford, Blackwell, 1994, 295–351.
- EHRENBERG A, My research in marketing: how it happened, Marketing Research, 2004, 16, 36–41.
- GREENE W., Functional forms for the negative binomial model for count data, Economics Letters, 2008, 99, 585–590.
- GUO J.Q., TRIVEDI P., Flexible parametric models for long-tailed patent count distributions, Oxford Bulletin of Economics and Statistics, 2002, 64, 63–82.
- JOHNSON N.L., KEMP A.W., KOTZ S., Univariate discrete distributions, Hoboken, Wiley, 2005.
- MCLACHLAN G.J., KRISHNAN T., The EM algorithm and extensions, Wiley, New York, 2008.
- MCLACHLAN G.J., PEEL D., Finite mixture models, Wiley, New York, 2000.
- PAWITAN Y., In all likelihood: statistical modelling and inference using likelihood. Oxford University Press, Oxford, 2001.
- R Development Core Team. R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2009. URL http://www.R-project.org.
- ROSS S.M., Introduction to probability models, Academic Press, Amsterdam, 2007.
- SELF S.G., LIANG K.Y., Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions, Journal of the American Statistical Association, 1987, 82, 604–610.
- TEICHER H., On the mixture of distributions, The Annals of Mathematical Statistics, 31, 1960, 55–73.
- WEDEL M., DESARBO W.S., BULT J.R., RAMASWAMY V., Latent class Poisson regression model for heterogeneous count data, Journal of Applied Econometrics, 8, 1993, 397–411.
- WINKELMANN R., Econometric analysis of count data, Springer, Berlin, 2008.
- WU C.F.J., On the Convergence properties of the EM Algorithm, The Annals of Statistics, 1983, 11, 95–103.
- YAKOWITZ S.J., SPRAGINS J.D., On the identifiability of finite mixtures, The Annals of Mathematical Statistics, 1968, 39, 209–214.
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