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Generalized Method of Moments and Generalized Estimating Functions Based on Probability Generating Function for Count Models
école d’actuariat, Université Laval, Ste Foy, Québec, Canada
- 1 école d’actuariat, Université Laval, Ste Foy, Québec, Canada
Open Journal of Statistics·Volume 10 (2020)·Pages 516–539·Published 8 May 2020·DOI10.4236/ojs.2020.103031
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Abstract
Generalized method of moments based on probability generating function is considered. Estimation and model testing are unified using this approach which also leads to distribution free chi-square tests. The estimation methods developed are also related to estimation methods based on generalized estimating equations but with the advantage of having statistics for model testing. The methods proposed overcome numerical problems often encountered when the probability mass functions have no closed forms which prevent the use of maximum likelihood (ML) procedures and in general, ML procedures do not lead to distribution free model testing statistics.
KeywordsMixture DistributionsConsistent Chi-Square TestsInfinitely Divisible DistributionsMixture DistributionsDistribution Free Test StatisticsModel Testing
- Johnson, N.L., Kotz, S. and Kemp, A.W. (1992) Univariate Discrete Distributions. Second Edition, Wiley, New York.
- Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2019) Loss Models: From Data to Decisions. Fifth Edition, Wiley, New York.
- Christoph, G. and Schreiber, K. (1998) Discrete Stable Random Variables. Statistics and Probability Letters, 37, 243-247. https://doi.org/10.1016/S0167-7152(97)00123-5
- Luong, A., Bilodeau, C. and Blier-Wong, C. (2018) Simulated Minimum Hellinger Distance Inference Methods for Count Data. Open Journal of Statistics, 8, 187-219. https://doi.org/10.4236/ojs.2018.81012
- Doray, L.G., Jiang, S.M. and Luong, A. (2009) Some Simple Method of Estimation for the Parameters of the Discrete Stable Distribution with the Probability Generating Function. Communications in Statistics—Simulation and Computation, 38, 2004-2017. https://doi.org/10.1080/03610910903202089
- Carrasco, M. and Florens, J.-P. (2000) Generalization of GMM to a Continuum of Moment Conditions. Econometric Theory, 16, 797-834. https://doi.org/10.1017/S0266466600166010
- Carrasco, M. and Kotchoni, R. (2017) Efficient Estimation Using Characteristic Function. Econometric Theory, 33, 479-526. https://doi.org/10.1017/S0266466616000025
- Martin, V., Hurn, S. and Harris, D. (2013) Econometric Modelling with Time Series: Specification, Estimation and Testing. Cambridge University Press, Cambridge.
- Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press, Princeton.
- Godambe, V.P. and Thompson, M.E. (1989) An Extension of Quasi-Likelihood Estimation. Journal of Statistical Planning and Inference, 22, 137-152. https://doi.org/10.1016/0378-3758(89)90106-7
- Morton, R. (1981) Efficiency of Estimating Equations and the Use of Pivots. Biometrika, 68, 227-233. https://doi.org/10.1093/biomet/68.1.227
- Liang, K.Y. and Zeger, S.L. (1986) Longitudinal Data Analysis Using Generalized Linear Models. Biometrika, 73, 13-22. https://doi.org/10.1093/biomet/73.1.13
- Hansen, L. (1982) Large Sample Properties of Generalized Method of Moment. Econometrica, 50, 1029-1054. https://doi.org/10.2307/1912775
- Rueda, R. and O’Reilly, F. (1999) Tests of Fit for Discrete Distributions Based on the Probability Generating Function. Communication in Statistics—Simulation and Computation, 28, 259-274. https://doi.org/10.1080/03610919908813547