Improved inference for the generalized Pareto distribution under linear, power and exponential normalization

Mohareb Khaled, Osama ; Mohamed Barakat, Haroon ; Khalil Rakha, Nourhan

Kybernetika, Tome 58 (2022) no. 6, pp. 883-902

Abstract (VO)

We discuss three estimation methods: the method of moments, probability weighted moments, and L-moments for the scale parameter and the extreme value index in the generalized Pareto distribution under linear normalization. Moreover, we adapt these methods to use for the generalized Pareto distribution under power and exponential normalizations. A simulation study is conducted to compare the three methods on the three models and determine which is the best, which turned out to be the probability weighted moments. A new computational technique for improving fitting quality is proposed and tested on two real-world data sets using the probability weighted moments. We looked back at various maximal data sets that had previously been addressed in the literature and for which the generalized extreme value distribution under linear normalization had failed to adequately explain them. We use the suggested procedure to find good fits.

Détail

DOI : 10.14736/kyb-2022-6-0883

Classification : 62F03, 62F10
Keywords: generalized Pareto distribution; generalized extreme value distribution; method of moments; probability weighted moments; L-moments; linear-power-exponential normalization

Bibliographie

[1] Balkema, A. A., Haan, L. de: Residual life time at great age. Ann. Prob. 2 (1974), 5, 792-804. | DOI | MR

[2] Barakat, H. M., Nigm, E. H., Zaid, E. O. Abo: Asymptotic distributions of record values under exponential normalization. Bull. Belg. Math. Soc. Simon Stevin 26 (2019), 743-758. | DOI | MR

[3] Barakat, H. M., Nigm, E. M-, Alaswed, H. A.: The Hill estimators under power normalization. App. Math. Modell. 45 (2017), 813-822. | DOI | MR

[4] Barakat, H. M., Nigm, E. M., El-Adll, M. E.: Comparison between the rates of convergence of extremes under linear and under power normalization. Stat. Pap. 51 (2010), 1, 149-164. | DOI | MR | Zbl

[5] Barakat, H. M., Nigm, E. M., Khaled, O. M.: Extreme value modeling under power normalization. App. Math. Modell. 37 (2013), 10162-10169. | DOI | MR

[6] Barakat, H. M., Nigm, E. M., Khaled, O. M.: Statistical modeling of extremes under linear and power normalizations with applications to air pollutions. Kuwait J. Sci. 41 (2014), 1, 1-19. | MR

[7] Barakat, H. M., Nigm, E. M., Khaled, O. M.: Bootstrap method for central and intermediate order statistics under power normalization. Kybernetika 51 (2015), 923-932. | DOI | MR

[8] Barakat, H. M., Khaled, O. M., Rakha, N. Khalil: Modeling of extreme values via exponential normalization compared with linear and power normalization. Symmetry 12 (2020), 11, 1876. | DOI

[9] Barakat, H. M., Nigm, E. M., Khaled, O. M., Alaswed, H. A.: The counterparts of Hill estimators under power normalization. Special Issue J. App. Stat. Sci. (JASS) 22 (2016), 1-2, 87-98. | MR

[10] Barakat, H. M., Omar, A. R., Khaled, O. M.: A new flexible extreme value model for modeling the extreme value data, with an application to environmental data. Stat. Probab. Lett. 130 (2017), 25-31. | DOI | MR

[11] Barakat, H. M., Nigm, E. M., Khaled, O. M., Alaswed, H. A.: The estimations under power normalization for the tail index, with comparison. AStA Adv. Stat. Anal. 102 (2018), 3, 431-454. | DOI | MR

[12] Barakat, H. M., Nigm, E. M., Khaled, O. M., Khan, F. M.: Bootstrap order statistics and modeling study of the air pollution. Comm. Statis.-Sim. Comp. 44 (2015), 1477-1491. | DOI | MR

[13] BuHamra, S., Al-Kandari, N., Al-Harbi, M.: Parametric and nonparametric bootstrap: An analysis of indoor air data from Kuwait. Kuwait J. Sci. 45 (2018), 2, 22-29.

[14] Castillo, E., Hadi, A. S., Balakrishnan, N., Sarabia, J. M.: Extreme Value and Related Models with Applications in Engineering and Science. Wiley, New-Jersey 2004. | MR

[15] Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer, London 2001. | MR

[16] Christoph, G., Falk, M.: A note on domains of attraction of p-max stable laws. Stat. Prob. Lett. 28 (1996), 279-284. | DOI | MR

[17] Davison, A. C., Smith, R. L.: Models for exceedances over high threshold. J. Roy. Stat. Soc. Ser. B. 52 (1990), 393-442. | MR

[18] Bermudez, P. de Zea, Kotz, S.: Parameter estimation of the generalized Pareto distribution - Part I. J. Stat. Plan. Inf. 140 (2010), 1353-1373. | DOI | MR

[19] Gnedenko, B. V.: Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44 (1943), 423-453. | DOI | MR

[20] Greenwood, J. A., Landwehr, J. M., Matalas, N. C., Wallis, J. R.: Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resour. Res. 15 (1979), 5, 1049-1054. | DOI

[21] Hosking, J. R. M.: L-moments: analysis and estimation of distributions using linear combinations of order statistics. J. Roy. Stat. Soc. B52 (1990), 105-124. | MR

[22] Landwehr, J. M., Matalas, N. C., Wallis, J. R.: Probability weighted moments compared with some traditional techniques for estimating Gumbel parameters and quantiles. Water Resour. Res. 15 (1979), 1055-1064. | DOI

[23] Nasri-Roudsari, D.: Limit distributions of generalized order statistics under power normalization. Comm. Statis.- Theory Meth. 28 (1999), 6, 1379-1389. | DOI | MR

[24] Pancheva, E.: Limit theorems for extreme order statistics under nonlinear normalization. In: Stability Problems for Stochastic Models, Lecture Notes in Math., Springer, Berlin 1155 (1984), 284-309. | MR

[25] Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3 (1975), 119-131. | MR

[26] Ravi, S., Mavitha, T. S.: New limit distributions for extreme under a nonlinear normalization. ProbStat Forum 9 (2016), 1-20.