On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution

Kouider Mohammed Ridha

doi:doi:10.11648/j.sjams.20190705.15

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On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution

Kouider Mohammed Ridha

Published in Science Journal of Applied Mathematics and Statistics (Volume 7, Issue 5)

Received: 14 April 2019 Accepted: 24 May 2019 Published: 26 October 2019

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Abstract

The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 7, Issue 5)
DOI	10.11648/j.sjams.20190705.15
Page(s)	89-94
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), 2019. Published by Science Publishing Group

Keywords

Extreme Value Index, Generalized Pareto Distributions, Excesses Over High Thresholds, Maximum Likelihood, The Modified Bisection Method Algorithm

References

[1]	Balkema, August A., and Laurens De Haan., 1974. “Residual life time at great age, The Annals of probability,”. 792-804.
[2]	Davison, Anthony C., 1984. “Behavior Modelling excesses over high thresholds, with an application. Statistical extremes and applications,”. Springer Netherlands, 461-482.
[3]	Drees, Holger, Ana Ferreira, and Laurens De Haan., 2004. “On maximum likelihood estimation of the extreme value index,” Annals of Applied Probability, 1179-1201.
[4]	Embrechts, P., and C. Klüppelberg., T. Mikosch., 1997. “Gas Modelling Extremal Events for Insurance and Finance, Applications of Mathematics-Stochastic Modelling and Applied Probability,”. Springer New York, No. 33.
[5]	Fisher, Ronald Aylmer, and Leonard Henry Caleb Tippett., 1928. “Limiting forms of the frequency distribution of the largest or smallest member of a sample,” Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, Vol. 24. No. 02.
[6]	Fachinger, Gnedenko, Boris., 1943. “Sur la distribution limite du terme maximum d'une serie aleatoire Annals of mathematics,”. 423-453.
[7]	Grimshaw, Scott D., 1993. “Computing maximum likelihood estimates for the generalized Pareto distribution,”. Technometrics 35.2, 185-191.
[8]	Haan, L. DE., 1970. “On regular variation and its application to the weak convergence of sample extremes,”. Mathematical Centre Tracts 32.
[9]	Haan, L de. and Ferreira, A., 2006. “Extreme Value Theory: An Introduction,” Springer.
[10]	Hosking, Jonathan RM, and James R. Wallis., 1987. “Parameter and quantile estimation for the generalized Pareto distribution,” Technometrics 29.3, 339-349.
[11]	Reiss, Rolf-Dieter, Michael Thomas, and R. D. Reiss., 2007. “Statistical analysis of extreme values,” Basel, Birkhüser, Vol. 2.
[12]	Smith, Richard L., 1985. “Computing Maximum likelihood estimation in a class of nonregular cases,” Biometrika 72.1, 67-90.
[13]	Tanakan, S., 2013. “A New Algorithm of Modified Bisection Method for Nonlinear Equation,” Applied Mathematical Sciences 7.123, 6107-6114.
[14]	Pickands III, James., 1975. “A Statistical inference using extreme order statistics,” the Annals of Statistics 119-131.

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Kouider Mohammed Ridha. (2019). On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution. Science Journal of Applied Mathematics and Statistics, 7(5), 89-94. https://doi.org/10.11648/j.sjams.20190705.15

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ACS Style

Kouider Mohammed Ridha. On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution. Sci. J. Appl. Math. Stat. 2019, 7(5), 89-94. doi: 10.11648/j.sjams.20190705.15

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AMA Style

Kouider Mohammed Ridha. On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution. Sci J Appl Math Stat. 2019;7(5):89-94. doi: 10.11648/j.sjams.20190705.15

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@article{10.11648/j.sjams.20190705.15,
  author = {Kouider Mohammed Ridha},
  title = {On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {7},
  number = {5},
  pages = {89-94},
  doi = {10.11648/j.sjams.20190705.15},
  url = {https://doi.org/10.11648/j.sjams.20190705.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20190705.15},
  abstract = {The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.},
 year = {2019}
}

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TY  - JOUR
T1  - On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution
AU  - Kouider Mohammed Ridha
Y1  - 2019/10/26
PY  - 2019
N1  - https://doi.org/10.11648/j.sjams.20190705.15
DO  - 10.11648/j.sjams.20190705.15
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  - 89
EP  - 94
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20190705.15
AB  - The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.
VL  - 7
IS  - 5
ER  -

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Author Information

Kouider Mohammed Ridha

Department of Mathematics, Applied Mathematics Laboratory, University of Mohamed Khider, Biskra, Algeria

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APA Style

Kouider Mohammed Ridha. (2019). On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution. Science Journal of Applied Mathematics and Statistics, 7(5), 89-94. https://doi.org/10.11648/j.sjams.20190705.15

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ACS Style

Kouider Mohammed Ridha. On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution. Sci. J. Appl. Math. Stat. 2019, 7(5), 89-94. doi: 10.11648/j.sjams.20190705.15

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AMA Style

Kouider Mohammed Ridha. On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution. Sci J Appl Math Stat. 2019;7(5):89-94. doi: 10.11648/j.sjams.20190705.15

Copy | Download

@article{10.11648/j.sjams.20190705.15,
  author = {Kouider Mohammed Ridha},
  title = {On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {7},
  number = {5},
  pages = {89-94},
  doi = {10.11648/j.sjams.20190705.15},
  url = {https://doi.org/10.11648/j.sjams.20190705.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20190705.15},
  abstract = {The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution
AU  - Kouider Mohammed Ridha
Y1  - 2019/10/26
PY  - 2019
N1  - https://doi.org/10.11648/j.sjams.20190705.15
DO  - 10.11648/j.sjams.20190705.15
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  - 89
EP  - 94
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20190705.15
AB  - The general Pareto distribution (GPD) has been widely used a lot in the extreme value for example to model exceedance over a threshold. Feature of The GPD that when applied to real data sets depends substantially and clearly on the parameter estimation process. Mostly the estimation is preferred by maximum likelihood because have a consistent estimator with lowest bias and variance. The objective of the present study is to develop efficient estimation methods for the maximum likelihood estimator for the shape parameter or extreme value index. Which based on the numerical methods for maximizing the log-likelihood by introduce an algorithm for computing maximum likelihood estimate of The GPD parameters. Finally, a numerical examples are given to illustrate the obtained results, they are carried out to investigate the behavior of the method.
VL  - 7
IS  - 5
ER  -

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