Brazilian Journal of Probability and Statistics

Improved inference for the generalized Pareto distribution

Juliana F. Pires, Audrey H. M. A. Cysneiros, and Francisco Cribari-Neto

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The generalized Pareto distribution is commonly used to model exceedances over a threshold. In this paper, we obtain adjustments to the generalized Pareto profile likelihood function using the likelihood function modifications proposed by Barndorff-Nielsen (Biometrika 70 (1983) 343–365), Cox and Reid (J. R. Stat. Soc. Ser. B. Stat. Methodol. 55 (1993) 467–471), Fraser and Reid (Utilitas Mathematica 47 (1995) 33–53), Fraser, Reid and Wu (Biometrika 86 (1999) 249–264) and Severini (Biometrika 86 (1999) 235–247). We consider inference on the generalized Pareto distribution shape parameter, the scale parameter being a nuisance parameter. Bootstrap-based testing inference is also considered. Monte Carlo simulation results on the finite sample performances of the usual profile maximum likelihood estimator and profile likelihood ratio test and also their modified versions is presented and discussed. The numerical evidence favors the modified profile maximum likelihood estimators and tests we propose. Finally, we consider two real datasets as illustrations.

Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 1 (2018), 69-85.

Dates
Received: March 2015
Accepted: August 2016
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1520046135

Digital Object Identifier
doi:10.1214/16-BJPS332

Mathematical Reviews number (MathSciNet)
MR3770864

Zentralblatt MATH identifier
06973949

Keywords
Bootstrap generalized Pareto distribution likelihood ratio test maximum likelihood estimation profile likelihood

Citation

Pires, Juliana F.; Cysneiros, Audrey H. M. A.; Cribari-Neto, Francisco. Improved inference for the generalized Pareto distribution. Braz. J. Probab. Stat. 32 (2018), no. 1, 69--85. doi:10.1214/16-BJPS332. https://projecteuclid.org/euclid.bjps/1520046135


Export citation

References

  • Barndorff-Nielsen, O. (1983). On a formula to the distribution of the maximum likelihood estimator. Biometrika 70, 343–365.
  • Castillo, E. and Hadi, A. S. (1997). Fitting the generalized Pareto distribution to data. Journal of the American Statistical Association 92, 1609–1620.
  • Castillo, E., Hadi, A. S., Balakrishnan, N. and Sarabia, J. M. (2005). Extreme Value and Related Models with Applications in Engineering and Science. Hoboken, NJ: Wiley.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. London: Springer.
  • Cox, D. and Hinkley, D. (1974). Theoretical Statistics. London: Chapman & Hall.
  • Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society. Series B 49, 1–39.
  • Cox, D. R. and Reid, N. (1993). A note on the calculation of adjusted profile likelihood. Journal of the Royal Statistical Society. Series B 55, 467–471.
  • Cox, D. R. and Snell, E. J. (1968). A general definition of residuals. Journal of the Royal Statistical Society. Series B 30, 248–275.
  • Davison, A. C. (2003). Statistical Models. Cambridge: Cambridge Univ. Press.
  • de Carvalho, M., Turkman, K. F. and Rua, A. (2013). Dynamic threshold modelling and the US business cycle. Journal of the Royal Statistical Society: Series C (Applied Statistics) 62, 535–550.
  • Doornik, J. A. (2013). An Object-Oriented Matrix Language Ox 7. London: Timberlake Consultants. Available at http://www.doornik.com/.
  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7, 1–26.
  • Ferreira, A. and Haan, L. (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli 20, 1717–1737.
  • Fraser, D. A. S. and Reid, N. (1995). Ancillaries and third-order significance. Utilitas Mathematica 47, 33–53.
  • Fraser, D. A. S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86, 249–264.
  • Giles, D. E., Feng, H. and Godwin, R. T. (2011). Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution. Econometrics Working Papers 1105, Department of Economics, University of Victoria.
  • Hosking, J. R. M. and Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29, 339–349.
  • Nocedal, J. and Wright, S. J. (2006). Numerical Optimization. New York: Springer.
  • Pace, L. and Salvan, A. (1997). Principles of Statistical Inference from a Neo-Fisherian Perspective. Singapore: World Scientific.
  • Pickands, J. (1975). Statistical inference using extreme order statistics. The Annals of Statistics 3, 119–131.
  • Severini, T. A. (1999). An empirical adjustment to the likelihood ratio statistic. Biometrika 86, 235–247.
  • Severini, T. A. (2000). Likelihood Methods in Statistics. Oxford: Oxford University Press.