On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution
Science Journal of Applied Mathematics and Statistics
Volume 7, Issue 5, October 2019, Pages: 89-94
Received: Apr. 14, 2019;
Accepted: May 24, 2019;
Published: Oct. 26, 2019
Views 355 Downloads 131
Kouider Mohammed Ridha, Department of Mathematics, Applied Mathematics Laboratory, University of Mohamed Khider, Biskra, Algeria
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.
Kouider Mohammed Ridha,
On Maximum Likelihood Estimates for the Shape Parameter of the Generalized Pareto Distribution, Science Journal of Applied Mathematics and Statistics.
Vol. 7, No. 5,
2019, pp. 89-94.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Balkema, August A., and Laurens De Haan., 1974. “Residual life time at great age, The Annals of probability,”. 792-804.
Davison, Anthony C., 1984. “Behavior Modelling excesses over high thresholds, with an application. Statistical extremes and applications,”. Springer Netherlands, 461-482.
Drees, Holger, Ana Ferreira, and Laurens De Haan., 2004. “On maximum likelihood estimation of the extreme value index,” Annals of Applied Probability, 1179-1201.
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.
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.
Fachinger, Gnedenko, Boris., 1943. “Sur la distribution limite du terme maximum d'une serie aleatoire Annals of mathematics,”. 423-453.
Grimshaw, Scott D., 1993. “Computing maximum likelihood estimates for the generalized Pareto distribution,”. Technometrics 35.2, 185-191.
Haan, L. DE., 1970. “On regular variation and its application to the weak convergence of sample extremes,”. Mathematical Centre Tracts 32.
Haan, L de. and Ferreira, A., 2006. “Extreme Value Theory: An Introduction,” Springer.
Hosking, Jonathan RM, and James R. Wallis., 1987. “Parameter and quantile estimation for the generalized Pareto distribution,” Technometrics 29.3, 339-349.
Reiss, Rolf-Dieter, Michael Thomas, and R. D. Reiss., 2007. “Statistical analysis of extreme values,” Basel, Birkhüser, Vol. 2.
Smith, Richard L., 1985. “Computing Maximum likelihood estimation in a class of nonregular cases,” Biometrika 72.1, 67-90.
Tanakan, S., 2013. “A New Algorithm of Modified Bisection Method for Nonlinear Equation,” Applied Mathematical Sciences 7.123, 6107-6114.
Pickands III, James., 1975. “A Statistical inference using extreme order statistics,” the Annals of Statistics 119-131.