Kybernetika 61 no. 4, 467-480, 2025

A regression method of estimation for generalized extreme value distribution

R. Anand and C. ChandranDOI: 10.14736/kyb-2025-4-0467

Abstract:

This study focuses on parameter estimation for the generalized extreme value distribution (GEVD) using the regression method described by \cite{van2012median}. A regression equation is derived from the cumulative distribution function and the scale parameter is estimated by applying the iterative re-weighted least squares in this regression equation. For estimating the shape parameter, a profile likelihood is constructed based on this regression equation. A comparison study of the regression method with other existing estimators derived from the method of moments, maximum likelihood, probability-weighted moments, l-moments, and maximum product spacing is performed for the GEVD. Also, the left truncated GEVD is considered and the behaviour of its hazard function is studied. The parameter estimates of the left truncated GEVD is also derived using the regression method. An extensive simulation study is conducted and the efficiencies of the estimation techniques are analysed. The bootstrap confidence intervals for the estimators are also constructed. Finally, a real data analysis is carried out to illustrate the applicability of the models and estimation techniques.

Keywords:

generalized extreme value distribution, regression method, Box-Cox transformation, profile likelihood

Classification:

60G70, 62F35

paper.pdf

References:

  1. B. A. A. Abdulali, M. A. Abu Bakar, K. Ibrahim and N. Mohd Ariff: Extreme value distributions: An overview of estimation and simulation. J. Probab. Statist. 1 (2022), 5449751.   DOI:10.1155/2022/5449751
  2. M. A. A. Bakar, N. M. Ariff and M. S. M. Nadzir: Comparative analysis between l-moments and maximum product spacing method for extreme pm concentration. In: International Conference on Mathematical Sciences and Statistics (ICMSS 2022), pp. 214-227.   CrossRef
  3. S. Baran, P. Szokol and M. Szabó: Truncated generalized extreme value distribution based emos model for calibration of wind speed ensemble forecasts. Environmetrics 32 (2021), 6, e2678.   DOI:10.1002/env.2678
  4. G. E. Box and D. R. Cox: An analysis of transformations. J. Royal Statist. Soc.: Series B (Methodological) 26 (1964), 2, 211-243.   DOI:10.1111/j.2517-6161.1964.tb00553.x
  5. A. Bücher and J. Segers: On the maximum likelihood estimator for the generalized extreme-value distribution. Extremes 20 (2017), 4, 839-872.   DOI:10.1007/s10687-017-0292-6
  6. S. Coles, J. Bawa, L. Trenner and P. Dorazio: An Introduction to Statistical Modeling of Extreme Values. Springer, London 2001.   CrossRef
  7. B. Efron: Bootstrap methods: Another look at the jackknife. Ann. Statist. 1 (1979), 1, 1-26.   DOI:10.1214/aos/1176344552
  8. B. Efron: Nonparametric standard errors and confidence intervals. Canadian J. Statist. 9 (1981), 2, 139-158.   DOI:10.2307/3314608
  9. B. Efron: Better bootstrap confidence intervals. J. American Statis. Assoc. 82 (1987), 397, 171-185.   DOI:10.1080/01621459.1987.10478410
  10. P. Embrechts, C. Klüppelberg and T. Mikosch: Modelling Extremal Events for Insurance and Finance. Springer, Berlin 1997.   CrossRef
  11. R. A. Fisher and L. H. C. Tippett: Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math. Proc. Cambridge Philosoph. Soc. 24 (1928), 2, 180-190.   DOI:10.1017/S0305004100015681
  12. M. Fréchet: Sur la loi de probabilité de l'écart maximum. Ann. Soc. Math. Polon. 6 (1927), 93-116.   CrossRef
  13. B. Gnedenko: Sur la distribution limite du terme maximum d'une serie aleatoire. Ann. Math. 44 (1943), 2, 423-453.   CrossRef
  14. J. R. Hosking: Algorithm as 215: Maximum-likelihood estimation of the parameters of the generalized extreme-value distribution. J. Royal Statist. Soc.: Series C (Applied Statistics) 34 (1985), 3, 301-310.   DOI:10.2307/2347484
  15. J. R. Hosking: L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. Royal Statist. Soc.: Series B (Methodological) 52 (1990), 1, 105-124.   DOI:10.1111/j.2517-6161.1990.tb01775.x
  16. J. R. M. Hosking, J. R. Wallis and E. F. Wood: Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27 (1985), 3, 251-261.   DOI:10.1080/00401706.1985.10488049
  17. A. F. Jenkinson: The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly J. Royal Meteorolog. Soc. 81 (1955), 348, 158-171.   DOI:10.1002/qj.49708134804
  18. M. R. Leadbetter, G. Lindgren and H. Rootzén: Extremes and Related Properties of Random Sequences and Processes. Springer, New York 1983.   CrossRef
  19. H. Madsen, P. F. Rasmussen and D. Rosbjerg: Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events: 1. At-site modeling. Water Resources Res. 33 (1997), 4, 747-757.   DOI:10.1029/96WR03848
  20. E. S. Martins and J. R. Stedinger: Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resources Res. 36 (2000), 3, 737-744.   DOI:10.1029/1999WR900330
  21. J. Pickands: Statistical inference using extreme order statistics. Ann. Statist. 3 (1975), 1, 119-131.   DOI:10.1214/aos/1176343003
  22. P. Prescott and A. Walden: Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika 67 (1980), 3, 723-724.   DOI:10.1093/biomet/67.3.723
  23. S. I. Resnick: Extreme Values, Regular Variation, and Point Processes. Springer, New York 1987.   CrossRef
  24. M. L. Rizzo: Statistical Computing with R. Chapman and Hall/CRC, New York 2019.   CrossRef
  25. R. L. Smith: Maximum likelihood estimation in a class of nonregular cases. Biometrika 72 (1985), 1, 67-90.   DOI:10.1093/biomet/72.1.67
  26. J. R. Stedinger, R. M. Vogel and E. Foufoula-Georgiou: Frequency analysis of extreme events. In: Handbook of Hydrology (D. R. Maidment, ed.), 18 (1993), McGraw-Hill, New York, pp. 18.1-18.66.   CrossRef
  27. J. M. Van Zyl: A median regression model to estimate the parameters of the three-parameter generalized pareto distribution. Commun. Statist.-Simul. Comput. 41 (2012), 4, 544-553.   DOI:10.1080/03610918.2011.595868
  28. R. Von Mises: La distribution de la plus grande de n valuers. Rev. Math. Union Interbalcanique 1 (1936), 141-160.   CrossRef
  29. T. Wong and W. Li: A note on the estimation of extreme value distributions using maximum product of spacings. Inst. Math. Statist. Lect. Notes - Monogr. Ser. 55 (2006), 272-283.   DOI:10.3323/jcorr.55.283
  30. A. Yılmaz, M. Kara and O. Özdemir: Comparison of different estimation methods for extreme value distribution. J. Appl. Statist. 48 (2021), 13-15, 2259-2284.   DOI:10.1080/02664763.2021.1940109
  31. A. Zellner: An Introduction to Bayesian Inference in Econometrics. Wiley, New York 1996.   CrossRef
  32. L. Zhang and B. A. Shaby: Uniqueness and global optimality of the maximum likelihood estimator for the generalized extreme value distribution. Biometrika 109 (2022), 3, 853-864.   DOI:10.1093/biomet/asab043