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Journal of Statistics and Management Systems

(Since 1998)
Hybrid ·Peer-reviewed·ISSN (Online): 2169-0014·ISSN (Print): 0972-0510

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Monthly Journal: Publishes peer-reviewed aticles on theoretical and applied statistics and management systems, expoloring industrial statistics, actuarial and decision sciences.

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Journal of Statistics and Management Systems
Open Access Research Article

A reduced-bias weighted least squares estimation of the extreme value index

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pp. 1499–1523Vol. 27Issue 8November 2024DOI: 10.47974/JSMS-981XML
Received:
02 Dec 2021
Accepted:
06 Jul 2022
Published Online:
18 Dec 2024
Article type:
Research Article
Language:
EN
Article no.:
JSMS-981
Pages:
1499–1523

Abstract

In this paper, we propose a reduced-bias estimator of the EVI for Pareto-type tails (heavy-tailed) distributions. This is derived using the weighted least squares method. It is shown that the estimator is asymptotically unbiased, asymptotically consistent and asymptotically normal under the second-order conditions on the underlying distribution of the data. The finite sample properties of the proposed estimator are studied through a simulation study. The results show that it is competitive to the existing estimators of the extreme value index in terms of bias and Mean Square Error. In addition, it yields estimates of g > 0  that are less sensitive to the number of top-order statistics, and hence, it alleviate the problem of selecting an optimal tail fraction to some extent. The proposed estimator is further illustrated using practical datasets from pedochemical and insurance.

Keywords

Extreme value theoryExtreme value indexWeighted least squaresLarge deviationsWeak law of large numbersLimit theorems

Subject Classifications

(2020) 62E2062G3262J0560F10

References

[1] J. Beirlant, G. Dierckx, Y. Goegebeur, and G. Matthys, ”Extreme value index estimation and an exponential regression model,” Extremes, vol. 2, no. 2, pp. 177–200 (1999).
[2] J. Beirlant, G. Dierckx, and A. Guillou, ”Estimation of the extreme-value index and generalized quantile plots,” Bernoulli, vol. 11, no. 6, pp. 949–970 (2005).
[3] J. Beirlant, G. Dierckx, A. Guillou, and C. Starica, ”On exponential representations of log-spacings of extreme order statistics,” Extremes, vol. 5, pp. 157–180 (2002).
[4] J. Beirlant, Y. Goegebeur, J. Teugels, and J. Segers, Statistics of Extremes: Theory and Applications, vol. 558, John Wiley & Sons (2004).
[5] J. Beirlant, G. Maribe, and A. Verster, ”Using shrinkage estimators to reduce bias and MSE estimation of heavy tails,” Statistical Journal, vol. 17, no. 1, pp. 91–108 (2019).
[6] J. Beirlant, P. Vynckier, and J. L. Teugels, ”Extreme value index estimation, Pareto quantile plots regression diagnostics,” Journal of the American Statistical Association, vol. 91, no. 436, pp. 1659–1667 (1996).
[7] P. Billingsley, Probability and Measure, 3rd ed., vol. 26, Wiley Series in Probability and Mathematical Statistics, New York (1983).
[8] M. Bladt, H. Albrecher, and J. Beirlant, ”Trimmed extreme value estimators for censored heavy-tailed data,” Electronic Journal of Statistics, vol. 15, no. 1, pp. 3112–3136 (2021).
[9] S. Buitendag, J. Beirlant, and T. de Wet, ”Ridge regression estimators for the extreme value index,” Extremes, vol. 22, no. 2, pp. 271–292 (2018).
[10] I. Cabral, F. Caeiro, and M. I. Gomes, ”On the comparison of several classical estimators of the extreme value index,” Communications in Statistics - Theory and Methods, pp. 1–18 (2020).
[11] F. Caeiro, M. I. Gomes, and D. Pestana, ”Direct reduction of bias of the classical Hill estimator,” Statistical, vol. 3, no. 2, pp. 113–136 (2005).
[12] L. De Haan and S. Resnick, ”On asymptotic normality of the Hill estimator,” Stochastic Models, vol. 14, no. 4, pp. 849–866 (1998).
[13] A. L. M. Dekkers, J. H. J. Einmahl, and L. de Haan, ”A moment estimator for the index of an extreme-value distribution,” The Annals of Statistics, vol. 17, no. 4, pp. 1833–1855 (1989).
[14] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, vol. 38, Springer Science & Business Media (2009).
[15] S. Elvidge and M. J. Angling, ”Using extreme value theory for determining the probability of Carrington-like solar flares,” Space Weather, vol. 16, no. 4, pp. 417–421 (2018).
[16] P. Embrechts, ”Extremes in economics and the economics of extremes,” in Extreme Values in Finance, Telecommunications, and the Environment, pp. 188–202, Chapman and Hall/CRC (2003).
[17] A. Feuerverger and P. Hall, ”Estimating a tail exponent by modelling departure from a Pareto distribution,” The Annals of Statistics, vol. 27, no. 2, pp. 760–781 (1999).
[18] B. Finkenstadt and H. Rootz´en, eds., Extreme Values in Finance, Telecommunications, and the Environment, Chapman & Hall/CRC (2004).
[19] M. Fraga Alves, L. de Haan, and T. Lin, ”Estimation of the parameter controlling the speed of convergence in extreme value theory,” Mathematical Methods of Statistics, vol. 12, no. 2, pp. 155–176 (2003).
[20] K. Gkillas and P. Katsiampa, ”An application of extreme value theory to cryptocurrencies,” Economics Letters, vol. 164, pp. 109–111 (2018).
[21] M. I. Gomes, L. De Haan, and L. H. Rodrigues, ”Extreme value index estimation for heavy-tailed models: Accommodation of bias in weighted log-excesses,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 70, no. 1, pp. 31–52 (2008).
[22] M. I. Gomes, U. D. Lisboa, M. Joa, and U. Te, ”Asymptotically unbiased estimators of the extreme value index based on external estimation of the second order parameter,” Extremes, pp. 5–31 (2002).
[23] M. B. Hill, ”A simple general approach to inference about the tail of a distribution,” Annals of Statistics, vol. 3, no. 5, pp. 1163–1174 (1975).
[24] R. Huisman, K. G. Kool, C. J. M. Koedijk, and F. Palm, ”Tail-index estimates in small samples,” Journal of Business & Economic Statistics, vol. 19, no. 2, pp. 208–216 (2001).
[25] J. H¨usler, D. Li, and S. M¨uller, ”Weighted least squares estimation of the extreme value index,” Statistics & Probability Letters, vol. 76, no. 9, pp. 920–930 (2006).
[26] M. M. Kithinji, P. N. Mwita, and A. O. Kube, ”Adjusted extreme conditional quantile autoregression with application to risk measurement,” Journal of Probability and Statistics, vol. 2021 (2021).
[27] F. Longin, ed., Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications, John Wiley & Sons (2016).
[28] V. Marimoutou, B. Raggad, and A. Trabelsi, ”Extreme value theory and value at risk: Application to oil market,” Energy Economics, vol. 31, no. 4, pp. 519–530 (2009).
[29] D. M. Mason, ”Laws of large numbers for sums of extreme values,” The Annals of Probability, pp. 754–764 (1982).
[30] N. Mehrnia and S. Coleri, ”Wireless channel modeling based on extreme value theory for ultra-reliable communications,” IEEE Transactions on Wireless Communications, vol. 21, no. 2, pp. 1064–1076 (2021).
[31] R. Minkah, ”Extreme value index estimation of the generalized Pareto distribution using a pivot from a transformed Pareto distribution,” Science and Development Journal, vol. 4, no. 1, pp. 1–19 (2020).
[32] R. Minkah, T. de Wet, and A. Ghosh, ”Robust estimation of Pareto-type extreme value index through an exponential regression model,” Communications in Statistics - Theory and Methods, pp. 1–19 (2021).
[33] R. Minkah, T. Doku-Amponsah, and D. Tertius, ”On extreme value index estimation under random censoring,” African Journal of Applied Statistics, vol. 5, no. 2, pp. 419–445 (2018).
[34] S. B. Muela, C. L. Mart´ın, and R. A. Sanz, ”An application of extreme value theory in estimating liquidity risk,” European Research on Management and Business Economics, vol. 23, no. 3, pp. 157–164 (2017).
[35] F. C. Onwuegbuche, S. B. Affognon, E. P. Enoc, M. O. Akinade, et al., ”Application of extreme value theory in predicting climate change induced extreme rainfall in Kenya,” International Journal of Statistics and Probability, vol. 8, no. 4, pp. 85–94 (2019).
[36] J. Pickands, ”Statistical inference using extreme order statistics,” The Annals of Statistics, vol. 3, no. 2, pp. 119–131 (1975).
[37] C. Rohrbeck, E. F. Eastoe, A. Frigessi, and J. A. Tawn, ”Extreme value modelling of water-related insurance claims,” The Annals of Applied Statistics, vol. 12, no. 1, pp. 246–282 (2018).
[38] L. Viharos, ”Weighted least-squares estimators of tail indices,” Probability and Mathematical Statistics-Wroclaw University, vol. 19, no. 2, pp. 249–265 (1999).
[39] C. Yozgatlgil and M. T¨urke¸s, ”Extreme value analysis and forecasting of maximum precipitation amounts in the western Black Sea subregion of Turkey,” International Journal of Climatology, vol. 38, no. 15, pp. 5447–5458 (2018).
[40] P. Hall, ”On some simple estimates of an exponent of regular variation,” Journal of the Royal Statistical Society: Series B, vol. 44, no. 1, pp. 37–42 (1982).

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