Abstract
Several methods have been used for estimating the parameters of the generalized Pareto distribution (GPD), namely maximum likelihood (ML), the method of moments (MOM) and the probability-weighted moments (PWM). It is known that for these estimators to exist, certain constraints have to be imposed on the range of the shape parameter,k, of the GPD. For instance, PWM and ML estimators only exist fork>−0.5 andk≤1, respectively. Moreover, and particularly for small sample sizes, the most efficient method to apply in any practical situation highly depends on a previous knowledge of the most likely values ofk. This clearly suggests the use of Bayesian techniques as a way of using prior information onk. In the present work, we address the issue of estimating the parameters of the GPD from a Bayesian point of view. The proposed approach is compared via a simulation study with ML, PWM and also with the elemental percentile method (EPM) which was developed by Castillo and Hadi (1997). The estimation procedure is then applied to two real data sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, B. C. andPress, S. J. (1989). Bayesian estimation and prediction for Pareto data.Journal of the American Statistical Association, 84:1079–1084.
Beirlant, J., Teugels, J. L., andVynckier, P. (1996).Practical analysis of extreme values. Leuven University Press, Leuven.
Best, N. G., Cowles, M. K., andVines, K. (1995). CODA—Convergence Diagnosis and Output Analysis software for Gibbs sampling output —version 0.30. Retrieved in http://www.mrc-bsu.cam.ac.uk.
Castillo, E. andHadi, A. S. (1997). Fitting the generalized Pareto distribution to data.Journal of the American Statistical Association, 92:1609–1620.
De Zea Bermudez, P. andAmaral Turkman, M. A. (2001). Bayesian approach to parameter estimation of the generalized Pareto distribution. Technical Report 7, CEAUL, University of Lisbon.
Dupuis, D. J. (1996). Estimating the probability of obtaining nonfeasible parameter estimates of the generalized Pareto distribution.Journal of Statistical Computation and Simulation, 54:197–209.
Gilks, W. R., Best, N. G., andTan, K. K. C. (1995). Adaptive rejection Metropolis sampling within Gibbs sampling.Applied Statistics, 44:455–472.
Grimshaw, S. D. (1993). Computing maximum likelihood estimates for the generalized Pareto distribution.Technometrics, 35:185–191.
Hosking, J. R. M. andWallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution.Technometrics, 29:339–349.
Kadane, J. B. (1980). Predictive and structural methods for eliciting prior distributions. In A. Zellner, ed.,Bayesian Analysis in Econometrics and Statistics, pp. 89–93. North-Holland, Amsterdam, 8th ed.
Pickands, J. (1975). Statistical inference using extreme order statistics.The Annals of Statistics, 3:199–131.
Rootzén, H. andTajvidi, N. (1997). Extreme value statistics and wind storm losses: A case study.Scandinavian Actuarial Journal, 1:70–94.
Smith, R. L. (1984). Threshold methods for sample extremes. In J. T. de Oliveira and D. Reidel, eds.,Statistical Extremes and Applications, pp. 621–638. North-Holland, Amsterdam.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Zea Bermudez, P., Turkman, M.A.A. Bayesian approach to parameter estimation of the generalized pareto distribution. Test 12, 259–277 (2003). https://doi.org/10.1007/BF02595822
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02595822
Key Words
- Elemental percentile method
- Gibbs sampling
- generalized Pareto distribution
- maximum likelihood
- peaks over threshold
- probability-weighted moments