# Some applications of probability generating function based methods to statistical estimation

Discussiones Mathematicae Probability and Statistics (2009)

- Volume: 29, Issue: 2, page 131-153
- ISSN: 1509-9423

## Access Full Article

top## Abstract

top## How to cite

topManuel L. Esquível. "Some applications of probability generating function based methods to statistical estimation." Discussiones Mathematicae Probability and Statistics 29.2 (2009): 131-153. <http://eudml.org/doc/277064>.

@article{ManuelL2009,

abstract = {After recalling previous work on probability generating functions for real valued random variables we extend to these random variables uniform laws of large numbers and functional limit theorem for the empirical probability generating function. We present an application to the study of continuous laws, namely, estimation of parameters of Gaussian, gamma and uniform laws by means of a minimum contrast estimator that uses the empirical probability generating function of the sample. We test the procedure by simulation and we prove the consistency of the estimator.},

author = {Manuel L. Esquível},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {probability generating function; empirical laws; estimation of parameters of continuous laws},

language = {eng},

number = {2},

pages = {131-153},

title = {Some applications of probability generating function based methods to statistical estimation},

url = {http://eudml.org/doc/277064},

volume = {29},

year = {2009},

}

TY - JOUR

AU - Manuel L. Esquível

TI - Some applications of probability generating function based methods to statistical estimation

JO - Discussiones Mathematicae Probability and Statistics

PY - 2009

VL - 29

IS - 2

SP - 131

EP - 153

AB - After recalling previous work on probability generating functions for real valued random variables we extend to these random variables uniform laws of large numbers and functional limit theorem for the empirical probability generating function. We present an application to the study of continuous laws, namely, estimation of parameters of Gaussian, gamma and uniform laws by means of a minimum contrast estimator that uses the empirical probability generating function of the sample. We test the procedure by simulation and we prove the consistency of the estimator.

LA - eng

KW - probability generating function; empirical laws; estimation of parameters of continuous laws

UR - http://eudml.org/doc/277064

ER -

## References

top- [1] D. Dacunha-Castelle and M. Duflo, Probabilités et Statistiques, Tome 2, Problèmes à temps mobile, Masson, Paris 1983. Zbl0535.62004
- [2] M.M. Dowling and M. Nakamura, Estimating parameters for discrete distributions via the empirical probability generating function, Commun. Stat., Simulation Comput 26 (1) (1997), 301-313. Zbl0902.62032
- [3] M.L. Esquível, Probability generating functions for discrete real-valued random variables, Teor. Veroyatn. Primen. 52 (1) (2007), 129-149; translation in Theory Probab. Appl. 52 (1) (2008), 40-57.
- [4] M.L. Esquível, Aplicações das funções geradoras de probabilidade a variáveis aleatórias reais, Proceedings of the XII Annual Congress Portuguese Statistical Society, Editors Carlos Braumann, Paulo Infante, Manuela Oliveira, Russell Alpizar Jara e Fernando Rosado (2005), 235-246.
- [5] A. Feuerverger, On the empirical saddlepoint approximation, Biometrika 76 (3) (1989), 457-464. Zbl0674.62019
- [6] A. Feuerverger and P. McDunnough, On statistical transform methods and their efficiency, The Can. J. of Stat. 12 (4) (1984), 303-317. Zbl0566.62021
- [7] P. Gaenssler, Empirical Processes, Lecture Notes-Monograph Series, volume 3, Institute of Mathematical Statistics, Hayward, CA 1983.
- [8] S. Kocherlakota and K. Kocherlakota, Goodness of fit tests for discrete distributions, Commun. Statist.-Theor. Meth. 15 (3) (1986), 815-829. Zbl0618.62049
- [9] P. Malliavin, Integration and Probability, Springer Verlag 1995.
- [10] M.S. Marques and V. Pérez-Abreu, Law of large numbers and central limit theorem for the empirical probability generating function of stationary random sequences and processes, Aportaciones Mat., Notas Invest. 4 (2) (1989), 100-109. Zbl0872.60018
- [11] M. Nakamura and V. Pérez-Abreu, Empirical probability generating function. An overview, Insur. Math. Econ. 12 (3) (1993), 349-366. Zbl0800.62075
- [12] M. Nakamura and V. Pérez-Abreu, Exploratory data analysis for counts using the empirical probability generating function, Commun. Stat., Theory Methods 22 (3) (1993), 827-842. Zbl0784.62001
- [13] M. Nakamura and V. Pérez-Abreu, Use of an empirical probability generating function for testing a Poisson model, Can. J. Stat. 21 (2) (1993), 149-156. Zbl0779.62039
- [14] W.R. Pestman, Mathematical Statistics, Walter de Gruyter, Paris, New York 1998. Zbl0911.62001
- [15] B.L.S. Prakasa Rao, Statistical Inference for Diffusion Type Processes, Arnold Hodder Headline Group 1999. Zbl0952.62077
- [16] B. Rémillard and R. Theodorescu, Inference based on the empirical probability generating function for mixtures of Poisson distributions, Stat. Decis. 18 (4) (2000), 349-366. Zbl1179.62046
- [17] R. Rueda and F. O'Reilly, Tests of fit for discrete distributions based on the probability generating function, Commun. Stat., Simulation Comput. 28 (1) (1999), 259-274. Zbl1054.62546
- [18] R. Rueda, V. Pérez-Abreu and F. O'Reilly, Goodness of fit for the Poisson distribution based on the probability generating function, Commun. Stat., Theory Methods 20 (10) (1991), 3093-3110. Zbl0800.62093

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.