On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples

Etienne Ouindllassida Jean Ouédraogo; Blaise Somé; Simplice Dossou-Gbété

doi:doi:10.11648/j.sjams.20170504.14

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On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples

Etienne Ouindllassida Jean Ouédraogo, Blaise Somé, Simplice Dossou-Gbété

Published in Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 4)

Received: 30 May 2017 Accepted: 14 June 2017 Published: 24 July 2017

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Abstract

This paper deals with a Maximum likelihood method to fit a three-parameter gamma distribution to data from an independent and identically distributed scheme of sampling. The likelihood hinges on the joint distribution of the n − 1 largest order statistics and its maximization is done by resorting to a MM-algorithm. Monte Carlo simulations is performed in order to examine the behavior of the bias and the root mean square error of the proposed estimator. The performances of the proposed method is compared to those of two alternatives methods recently available in the literature: the location and scale parameters free maximum likelihood estimators (LSPF-MLE) of Nagatsuka & al. (2014), and Bayesian Likelihood (BL) method of Hall and Wang (2005). As in several papers on the three-parameter gamma fitting (Cohen and Whitten (1986), Tzavelas (2009), Nagatsuka & al. (2014), etc.), the classical dataset on the maximum flood levels data in millions of cubic feet per second for the Susquehanna River at Harrisburg, Pennsylvania, over 20 four-year periods from 1890–1969 from Antle and Dumonceaux’s paper (1973) is consider to illustrate the proposed method.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 4)
DOI	10.11648/j.sjams.20170504.14
Page(s)	147-163
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Estimation, Likelihood, MM-algorithm, Order Statistics, Pearson Type III Model, Three-Parameter Gamma Model, Left Censoring

References

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[3]	H. Nagatsuka, N. Balakrishnan, T. Kamakura: “A consistent Method of Estimation For the Three-Parameter Gamma Distribution”, Communications in Statistics-Theory and Methods, pp. 3905-3926, 2014.
[4]	K. O. and Shenton Bowman: “Problems with maximum likelihood estimation and the 3 parameter gamma distribution”, Journal of Statistical Computation and Simulation, pp. 391-401, 2002.
[5]	R. C. H. Cheng, T. C. Iles: “Embedded models in three-parameter distributions and their estimation”, Journal of the Royal Statistical Society. Series B (Methodological), pp. 135-149, 1990.
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[8]	G. Tzavelas: “Sufficient conditions for the existence of a solution for the log-likelihood equations in three-parameter gamma distribution”, Communications in Statistics-theory and methods, pp. 1371-1382, 2008.
[9]	N. Balakrishnan, J. Wang: “Simple efficient estimation for the three-parameter gamma distribution”, Journal of statistcal planning and inference, pp. 115-126, 2000.
[10]	P. Hall, J. Z. Wang: “Bayesian Likelihood Methods for estimating the end point of a Distribution”, Journal of the Royal Statistical Society, pp. 717-729, 2005.
[11]	G. Tzavelas: “Maximum likelihood parameter estimation in the three-parameter gamma distribution with use of Mathematica”, Journal of Statistical Computation and Simulation, pp. 1457-1466, 2009.
[12]	K. and Hunter Lange, I. Yang: “Optimization transfer using surrogate objective functions (with discussion)”, Journal of Computational and Graphical Statistics, pp. 1-20, 2000.
[13]	D. R. Hunter, K. Lange: “A tutorial on MM algorithms”, The American Statistician, pp. 30-37, 2004.
[14]	K. Lange: Optimization, Second Edition. Springer New York, 2013.
[15]	W. C. F. Jeff: “On the Convergence Properties of the EM Algorithm”, The Annals of Statistics, pp. 95-103, 1983.
[16]	F. Vaida: “Parameter convergence for EM and MM algorithms”, Statist. Sinica, pp. 831-840, 2005.
[17]	R Core Team: R: A Language and Environment for Statistical Computing. http://www.R-project.org/, 2015.
[18]	M. Becker, S. Klößner: PearsonDS: Pearson Distribution System.. http://CRAN.R-project.org/package=PearsonDS, R package version 0.97, 2013.
[19]	R. Dumonceaux, C. E. Antle: “Discriminations between the log-normal and the weibull distributions”, Technometrics, pp. 923-926, 1973.
[20]	H. Hirose: “Maximum likelihood parameter estimation in the three-parameter gamma distribution”, Computational Statistics and Data Analysis, pp. 343-354, 1995.
[21]	A. C. Cohen, B. J. Whitten: “Modified moment estimation for the three-parameter gamma distribution”, Journal of Quality Technology, pp. 53-62, 1986.
[22]	J. Carpenter, J. Bithell: “Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians.”, Statistics in Medecine, pp. 1141-1164, 2000.

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APA Style

Etienne Ouindllassida Jean Ouédraogo, Blaise Somé, Simplice Dossou-Gbété. (2017). On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples. Science Journal of Applied Mathematics and Statistics, 5(4), 147-163. https://doi.org/10.11648/j.sjams.20170504.14

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ACS Style

Etienne Ouindllassida Jean Ouédraogo; Blaise Somé; Simplice Dossou-Gbété. On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples. Sci. J. Appl. Math. Stat. 2017, 5(4), 147-163. doi: 10.11648/j.sjams.20170504.14

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AMA Style

Etienne Ouindllassida Jean Ouédraogo, Blaise Somé, Simplice Dossou-Gbété. On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples. Sci J Appl Math Stat. 2017;5(4):147-163. doi: 10.11648/j.sjams.20170504.14

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@article{10.11648/j.sjams.20170504.14,
  author = {Etienne Ouindllassida Jean Ouédraogo and Blaise Somé and Simplice Dossou-Gbété},
  title = {On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {5},
  number = {4},
  pages = {147-163},
  doi = {10.11648/j.sjams.20170504.14},
  url = {https://doi.org/10.11648/j.sjams.20170504.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20170504.14},
  abstract = {This paper deals with a Maximum likelihood method to fit a three-parameter gamma distribution to data from an independent and identically distributed scheme of sampling. The likelihood hinges on the joint distribution of the n − 1 largest order statistics and its maximization is done by resorting to a MM-algorithm. Monte Carlo simulations is performed in order to examine the behavior of the bias and the root mean square error of the proposed estimator. The performances of the proposed method is compared to those of two alternatives methods recently available in the literature: the location and scale parameters free maximum likelihood estimators (LSPF-MLE) of Nagatsuka & al. (2014), and Bayesian Likelihood (BL) method of Hall and Wang (2005). As in several papers on the three-parameter gamma fitting (Cohen and Whitten (1986), Tzavelas (2009), Nagatsuka & al. (2014), etc.), the classical dataset on the maximum flood levels data in millions of cubic feet per second for the Susquehanna River at Harrisburg, Pennsylvania, over 20 four-year periods from 1890–1969 from Antle and Dumonceaux’s paper (1973) is consider to illustrate the proposed method.},
 year = {2017}
}

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TY  - JOUR
T1  - On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples
AU  - Etienne Ouindllassida Jean Ouédraogo
AU  - Blaise Somé
AU  - Simplice Dossou-Gbété
Y1  - 2017/07/24
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170504.14
DO  - 10.11648/j.sjams.20170504.14
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
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EP  - 163
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170504.14
AB  - This paper deals with a Maximum likelihood method to fit a three-parameter gamma distribution to data from an independent and identically distributed scheme of sampling. The likelihood hinges on the joint distribution of the n − 1 largest order statistics and its maximization is done by resorting to a MM-algorithm. Monte Carlo simulations is performed in order to examine the behavior of the bias and the root mean square error of the proposed estimator. The performances of the proposed method is compared to those of two alternatives methods recently available in the literature: the location and scale parameters free maximum likelihood estimators (LSPF-MLE) of Nagatsuka & al. (2014), and Bayesian Likelihood (BL) method of Hall and Wang (2005). As in several papers on the three-parameter gamma fitting (Cohen and Whitten (1986), Tzavelas (2009), Nagatsuka & al. (2014), etc.), the classical dataset on the maximum flood levels data in millions of cubic feet per second for the Susquehanna River at Harrisburg, Pennsylvania, over 20 four-year periods from 1890–1969 from Antle and Dumonceaux’s paper (1973) is consider to illustrate the proposed method.
VL  - 5
IS  - 4
ER  -

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Author Information

Etienne Ouindllassida Jean Ouédraogo

Laboratory of Numerical Analysis, Informatics and Bio-mathematics (LANIBIO), Unit of Formation and Research in Exact and Applied Sciences, University of Ouagadougou, Ouagadougou, Burkina Faso
Blaise Somé

Laboratory of Numerical Analysis, Informatics and Bio-mathematics (LANIBIO), Unit of Formation and Research in Exact and Applied Sciences, University of Ouagadougou, Ouagadougou, Burkina Faso
Simplice Dossou-Gbété

Laboratory of Mathematics and Their Applications of Pau (LMAP), University of Pau and Pays de l’Adour, Pau, France

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APA Style

Etienne Ouindllassida Jean Ouédraogo, Blaise Somé, Simplice Dossou-Gbété. (2017). On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples. Science Journal of Applied Mathematics and Statistics, 5(4), 147-163. https://doi.org/10.11648/j.sjams.20170504.14

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ACS Style

Etienne Ouindllassida Jean Ouédraogo; Blaise Somé; Simplice Dossou-Gbété. On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples. Sci. J. Appl. Math. Stat. 2017, 5(4), 147-163. doi: 10.11648/j.sjams.20170504.14

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AMA Style

Etienne Ouindllassida Jean Ouédraogo, Blaise Somé, Simplice Dossou-Gbété. On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples. Sci J Appl Math Stat. 2017;5(4):147-163. doi: 10.11648/j.sjams.20170504.14

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@article{10.11648/j.sjams.20170504.14,
  author = {Etienne Ouindllassida Jean Ouédraogo and Blaise Somé and Simplice Dossou-Gbété},
  title = {On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {5},
  number = {4},
  pages = {147-163},
  doi = {10.11648/j.sjams.20170504.14},
  url = {https://doi.org/10.11648/j.sjams.20170504.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20170504.14},
  abstract = {This paper deals with a Maximum likelihood method to fit a three-parameter gamma distribution to data from an independent and identically distributed scheme of sampling. The likelihood hinges on the joint distribution of the n − 1 largest order statistics and its maximization is done by resorting to a MM-algorithm. Monte Carlo simulations is performed in order to examine the behavior of the bias and the root mean square error of the proposed estimator. The performances of the proposed method is compared to those of two alternatives methods recently available in the literature: the location and scale parameters free maximum likelihood estimators (LSPF-MLE) of Nagatsuka & al. (2014), and Bayesian Likelihood (BL) method of Hall and Wang (2005). As in several papers on the three-parameter gamma fitting (Cohen and Whitten (1986), Tzavelas (2009), Nagatsuka & al. (2014), etc.), the classical dataset on the maximum flood levels data in millions of cubic feet per second for the Susquehanna River at Harrisburg, Pennsylvania, over 20 four-year periods from 1890–1969 from Antle and Dumonceaux’s paper (1973) is consider to illustrate the proposed method.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - On Maximum Likelihood Estimation for the Three Parameter Gamma Distribution Based on Left Censored Samples
AU  - Etienne Ouindllassida Jean Ouédraogo
AU  - Blaise Somé
AU  - Simplice Dossou-Gbété
Y1  - 2017/07/24
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170504.14
DO  - 10.11648/j.sjams.20170504.14
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 147
EP  - 163
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170504.14
AB  - This paper deals with a Maximum likelihood method to fit a three-parameter gamma distribution to data from an independent and identically distributed scheme of sampling. The likelihood hinges on the joint distribution of the n − 1 largest order statistics and its maximization is done by resorting to a MM-algorithm. Monte Carlo simulations is performed in order to examine the behavior of the bias and the root mean square error of the proposed estimator. The performances of the proposed method is compared to those of two alternatives methods recently available in the literature: the location and scale parameters free maximum likelihood estimators (LSPF-MLE) of Nagatsuka & al. (2014), and Bayesian Likelihood (BL) method of Hall and Wang (2005). As in several papers on the three-parameter gamma fitting (Cohen and Whitten (1986), Tzavelas (2009), Nagatsuka & al. (2014), etc.), the classical dataset on the maximum flood levels data in millions of cubic feet per second for the Susquehanna River at Harrisburg, Pennsylvania, over 20 four-year periods from 1890–1969 from Antle and Dumonceaux’s paper (1973) is consider to illustrate the proposed method.
VL  - 5
IS  - 4
ER  -

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