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Journal of Statistical Theory and Practice

, Volume 9, Issue 1, pp 171–183 | Cite as

Estimation of the Shape Parameter of a Generalized Pareto Distribution Based on a Transformation to Pareto Distributed Variables

  • J. Martin van Zyl
Article

Abstract

Random variables of the generalized three-parameter Pareto distribution can be transformed to that of the Pareto distribution using an affine parameter-dependent transformation. Explicit expressions exist for the maximum likelihood estimators of the parameters of the Pareto distribution. The performance of the estimation of the shape parameter of generalized Pareto distributed using estimated parameters to perform the transformation is tested. It was found to improve the performance with respect to relative efficiency.

Keywords

Generalized Pareto Estimation Pareto Transformation 

AMS Subject Classification

62F10 62E17 

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References

  1. Balkema, A., and L. de Haan. 1974. Life time at great age. Ann. Probability, 2, 792–804.MathSciNetCrossRefGoogle Scholar
  2. Castillo, E. and A. S. Hadi. 1997. Fitting the generalized Pareto distribution to data. J. Am. Stat. Assoc., 92(440), 1609–1620.MathSciNetCrossRefGoogle Scholar
  3. De Zea Bermudez, P., and S. Kotz. 2010. Parameter estimation of the generalized Pareto distribution—Part I. J. Stat. Plan. Inference, 140(6), 1353–1373.CrossRefGoogle Scholar
  4. Del Castillo, J., and J. Daoudi. 2009. Estimation of the generalized Pareto distribution. Stat. Probability Lett., 79, 684–688.MathSciNetCrossRefGoogle Scholar
  5. Grimshaw, S. D. 1993. Computing maximum likelihood estimates for the generalized Pareto distributions. Technometrics, 35, 185–191.MathSciNetCrossRefGoogle Scholar
  6. Hill, B. M. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat., 3, 1163–1174.MathSciNetCrossRefGoogle Scholar
  7. Hosking, J. R. M., J. R. Wallis, and E. F. Wood. 1985. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27(3), 251–261.MathSciNetCrossRefGoogle Scholar
  8. Hosking, J. R. M. 1986. The theory of probability weighted moments. Research report RC12210. Yorktown Heights, NY: IBM Research Division.Google Scholar
  9. Hosking, J. R. M. 1990. L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. R. Stat. Society Ser. B, 52, 105–124.MathSciNetzbMATHGoogle Scholar
  10. Johnson, N. L., S. Kotz, and N. Balakrishnan. 1994. Continuous univariate distributions, vol. 1. New York, NY: Wiley.zbMATHGoogle Scholar
  11. Koutrouvelis, I. A. 1980. Regression-type estimation of the parameters of stable laws. J. Am. Stat. Assoc., 75, 918–928.MathSciNetCrossRefGoogle Scholar
  12. Landwehr, J. M., N. C. Matalas, and J. R. Wallis. 1979. Probability weighted moments compared with some traditional techniques for estimating Gumbel parameters and quantiles. Water. Resources Res., 15, 1055–1064.CrossRefGoogle Scholar
  13. Mackay, E. B. L., P. G. Challenor, and A. S. Bahaj. 2011. A comparison of estimators for the generalised Pareto distribution. Ocean Eng., 38, 1338–1346.CrossRefGoogle Scholar
  14. Markovich, N. 2007. Nonparametric analysis of univariate heavy-tailed data: Research and practice. New York, NY: Wiley.CrossRefGoogle Scholar
  15. McCulloch, J. H. 1997. Measuring tail thickness to estimate the stable index α: A critique. J. Am. Stat. Assoc., 15(1), 74–81.MathSciNetGoogle Scholar
  16. McNeil, A. J., and T. Saladin. 1997. The peaks over thresholds method for estimating high quantiles of loss distributions. In Proceeding of the 28th International ASTIN Colloquium (Cairns, Australia), Casual Actuarial Society, Arlington, VA, 22–43.Google Scholar
  17. Meintanis, S. G., and Y. Bassiakos. 2007. Data-transformation and test of fit for the generalized Pareto hypothesis. Commun. Stat. Theory Methods, 36(4), 833–849.MathSciNetCrossRefGoogle Scholar
  18. Pickands, J. 1975. Statistical inference using extreme order statistics. Ann. Stat., 3, 119–131.MathSciNetCrossRefGoogle Scholar
  19. Paulson, A. S., W. E. Holcomb, and R. A. Leitch. 1975. The estimation of the parameters of the stable laws. Biometrika, 62, 163–170.MathSciNetCrossRefGoogle Scholar
  20. Reiss, R. D., and U. Cormann. 2008. An example of real-life data where the Hill estimator is correct. In Advances in mathematical and statistical modeling, ed. B. C. Arnold et al. Boston, MA: Birkhäuser, 209–216.CrossRefGoogle Scholar
  21. Smith, R. L. 1984. Threshold methods for sample extremes. In Statistical extremes and applications, ed. J. Tiago de Oliveira, 621–638. Dordrecht, The Netherlands: Reidel.CrossRefGoogle Scholar
  22. Zhang, J., and M. A. Stephens. 2009. A new and efficient estimation method for the generalized Pareto distribution. Technometrics, 51(3), 316–325.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  1. 1.Department of Mathematical Statistics and Actuarial ScienceUniversity of the Free StateBloemfonteinSouth Africa

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