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Minimum distance estimators for count data based on the probability generating function with applications

Abstract

This paper studies properties of parameter estimators obtained by minimizing a distance between the empirical probability generating function and the probability generating function of a model for count data. Specifically, it is shown that, under certain not restrictive conditions, the resulting estimators are consistent and, suitably normalized, asymptotically normal. These properties hold even if the model is misspecified. Three applications of the obtained results are considered. First, we revisit the goodness-of-fit problem for count data and propose a weighted bootstrap estimator of the null distribution of test statistics based on the above cited distance. Second, we give a probability generating function version of the model selection test problem for separate, overlapping and nested families of distributions. Finally, we provide an application to the problem of testing for separate families of distributions. All applications are illustrated with numerical examples.

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Acknowledgements

The authors thank the anonymous referees for their constructive comments and suggestions which helped to improve the presentation. M.D. Jiménez-Gamero acknowledges financial support from Grant MTM2014-55966-P of the Spanish Ministry of Economy and Competitiveness.

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Correspondence to A. Batsidis.

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Jiménez-Gamero, M.D., Batsidis, A. Minimum distance estimators for count data based on the probability generating function with applications. Metrika 80, 503–545 (2017). https://doi.org/10.1007/s00184-017-0614-3

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Keywords

  • Probability generating function
  • Consistency
  • Asymptotic normality
  • Goodness-of-fit
  • Model selection
  • Testing for separate families