In general, the classical methods used to estimate the quantiles of extreme random variables are maximum likelihood and the computation of moments.

In this paper, we estimate Gumbel density parameters with minimum Hellinger distance estimator (MHDE) by implementing a quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) numerical algorithm. Also, we determine the kernel density adapted to the Gumbel density. One of the important problems of the minimum Hellinger distance estimator (MHDE) resides in the choice of the kernel density. The results showed that the Parzen-Rosenblatt kernel density is closer to the Gumbel density. The bias of the quantiles estimated by the MHDE is found to be lower compared to those of the maximum likelihood estimator (MLE) in the case of real data from the L’Ardières station of Beaujeu.

Gumbel distribution, kernel density, Hellinger distance, quasi-Newton, estimation quantiles.

Received: March 24, 2022; Accepted: May 11, 2022; Published: July 4, 2022

How to cite this article: Kolé Keita, Gueï Cyrille Okou, Aubin Yao N’Dri and Ouagnina Hili, Estimation quantiles of the Gumbel distribution based on the Hellinger distance: application of data from L’Ardières station of Beaujeu (Rhône Department), Far East Journal of Theoretical Statistics 65 (2022), 71-95. http://dx.doi.org/10.17654/0972086322007

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] M. Belbachir, G. C. Okou and B. Labloul, Normalisation functions and the reliability of the estimation of the GEV return level: a comparative study based on CAC 40 index, International Journal of Statistics and Economics 18(4) (2017), 122 139.
[2] A. Beran, Minimum Hellinger distance estimates for parametric models, Quart. J. Roy. Meter. Soc. 5(3) (1977), 445-463.
[3] M. Déqué, Frequency of precipitation and temperature extremes over France in an anthropogenic scenario: model results and statistical correction according to observed values, Glob. and Plan. Change 57(1-2) (2007), 16-26.
[4] S. El Adlouni and T. B. M. J. Ouarda, Comparaison des méthodes d’estimation des paramètres du modèle gev non-stationnaire, Revue des Sciences de l’Eau 21(1) (2008), 35-50.
[5] S. El Adlouni, T. B. M. J. Ouarda and B. Bobée, Orthogonal projection l-moment estimators for three-parameter distributions, Advances and Applications in Statistics 7(2) (2007), 193-209.
[6] R. A. Fisher and L. H. C. Tippet, Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proceedings of the Cambridge Philosophical Society 24(2) (1928), 180-190.
[7] B. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Annals of Mathematics 44(3) (1943), 1-14.
[8] Jaya Bishwal, Financial extremes: a short review, Advances and Applications in Statistics 25(1) (2011), 1-14.
[9] O. Hili, Hellinger distance estimation of SSAR models, Statist. Probab. Lett. 53(3) (2001), 305-314.
[10] O. Hili, Hellinger distance estimation of nonlinear dynamical systems, Statist. Probab. Lett. 63(2) (2003), 177-184.
[11] O. Hili, Hellinger distance estimation of general bilinear time series models, Stat. Methodol. 5(2) (2008), 119-128.
[12] A. Jenkinson, The frequency distribution of the annual maximum (or minimum) of meteorological elements, Quart. J. Roy. Meter. Soc. 81(348) (1955), 158-171.
[13] A. R. Kadjo, O. Hili and A. N’dri, Minimum Hellinger distance estimation of multivariate GARCH processes, Far East Journal of Theoretical Statistics 54(6) (2018), 503-526.
[14] A. R. Kadjo, O. Hili and A. N’dri, Estimation and asymptotic properties of a stationary univariate GARCH process, Afrika Statistika 15(1) (2020), 2225 2246.
[15] E. J. Martins and J. R. Stedinger, Generalized maximum likelihood GEV quantile estimators for hydrologic data, Water Resour. Res. 36(3) (2000), 737-744.
[16] A. N’dri and O. Hili, Estimation par la distance de Hellinger des processus Gaussiens stationnaires fortement dependants, Comptes Rendus, Mathématique, Académie des Sciences, Paris 349(17-18) (2011), 991-994.
[17] G. C. Okou, Estimation du risque financier par l’approche de Peaks Over Threshold (POT) et de la théorie des copules, PhD thesis, Faculté des Sciences, Université Mohammed V-Agdal, 2014.
[18] G. C. Okou, K. Keita and A. Kamagaté, Estimation of extreme quantiles for non stationary Gumbel models in the case of non linear normalization, International Journal of Mathematics and Computation 30(4) (2019), 17-32.
[19] E. Parzen, On estimation of a probability density function and mode, Annals of Mathematical Statistics 33(3) (1962), 1065-1076.
[20] R. D. Reiss and M. Thomas, Statistical Analysis of Extreme Value with Applications to Assurance, Finance, Hydrology and Other Fields, Basel, Birkhauser, Verlag, 2001.
[21] M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Annals of Mathematical Statistics 27 (1956), 832-837.
[22] A. Sankarasubramanian and U. Lall, Flood quantiles in a changing climate: Seasonal forecasts and causal relations, Water Resour. Res. 39(5) (2003), 1134.
[23] J. A. Smith, G. N. Day and M. D. Kane, Nonparametric framework for long-range streamflow forecasting, J. Water Resour. Plann. Manage. 118(1) (1992), 82-91.
[24] R. Smith, Extreme value statistics in meteorology and the environment, Environmental Statistics, Chapter 8, NSF-CBMS Conference Notes, 2001.
[25] J. R. Stedinger, R. M. Vogel and E. Foufoula-Georgiou, Frequency analysis of extreme events, Handbook of Hydrology, McGraw-Hill, New York, 1993.
[26] M.-S. Tsai and L.-C. Chen, The calculation of capital requirement using extreme value theory, Economic Modelling 28 (2011), 390-395.
[27] X. Yingmei, D. Diawara and L. Kané, An application of extreme value theory in automobile insurance: a new approach to the convex combination of two variables thresholds minimizing the variance of this combination, Advances and Applications in Statistics 48(2) (2016), 91-108.
[28] A. Zoglat, S. El Adlouni, F. Badaoui, A. Amar and G. C. Okou, Statistical methods to expect extreme values: application of pot approach to cac40 return index, International Journal of Statistics and Economics 10(1) (2013), 1-13.
[29] A. Zoglat, S. El Adlouni, F. Badaoui, A. Amar and G. C. Okou, Managing hydrological risks with extreme modeling: application of peaks over threshold model to the Loukkos watershed, Morocco, Journal of Hydrologic Engineering 19(9) (2014), 16-26.