Abstract
Motivated by applications requiring quantile estimates for very small probabilities of exceedance p n ≪1/n, this article addresses estimation of high quantiles for p n satisfying \(p_{n}\in [n^{-\tau _{2}},n^{-\tau _{1}}]\) for some τ 1>1 and τ 2>τ 1. For this purpose, the tail regularity assumption logU∘ exp∈E R V (with U the left-continuous inverse of 1/(1−F), and ERV the extended regularly varying functions) is explored as an alternative to the classical regularity assumption U∈E R V (corresponding to the Generalised Pareto tail limit). Motivation for the alternative regularity assumption is provided, and it is shown to be equivalent to a limit relation for the logarithm of the survival function, the log-GW tail limit, which generalises the GW (Generalised Weibull) tail limit, a generalisation of the Weibull tail limit. The domain of attraction is described, and convergence results are presented for quantile approximation and for a simple quantile estimator based on the log-GW tail. Simulations are presented, and advantages and limitations of log-GW-based estimation of high quantiles are indicated.
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References
Adams, J., Atkinson, G.: Development of seismic hazard maps for the proposed 2005 edition of the National Building Code of Canada. Can. J. Civ. Eng. 30, 255–271 (2003)
Bingham, N.H., Goldie, C.M., Teugels, J.L: Regular variation. Cambridge Univ. Press (1987)
Bojanic, R., Seneta, E.: Slowly varying functions and asymptotic relations. JMAA 34, 302–315 (1971)
Broniatowski, M.: On the estimation of the Weibull tail coefficient. J. Stat. Plan. Inference 35, 349–366 (1993)
Cai, J.J., de Haan, L., Zhou, C.: Bias correction in extreme value statistics with index around zero. Extremes 16(2), 173–201 (2013)
Cope, E.W., Mignolia, G., Antonini, G., Ugoccioni, R.: Challenges and pitfalls in measuring operational risk from loss data. J. Oper. Risk 4(4), 3–27 (2009)
Dekkers, A.L.M., Einmahl, J.H.J., De Haan, L.: A Moment Estimator for the Index of an Extreme-value Distribution. Ann. Stat. 17(4), 1833–1855 (1989)
Diebolt, J., Gardes, L., Girard, S., Guillou, A.: Bias-reduced estimators of the Weibull tail-coefficient. Test 17, 311–331 (2008)
Drees, H.: Extreme quantile estimation for dependent data, with applications to finance. Bernoulli 9(4), 617–657 (2003)
Einmahl, J.H.J., Mason, D.: Strong limit theorems for weighted quantile processes. Ann. Prob. 4, 1623–1643 (1988)
Gardes, L., Girard, S., Guillou, A.: Weibull tail-distributions revisited: a new look at some tail estimators. J. Stat. Plan. Inference 141, 429–444 (2011)
Gardes, L., Girard, S.: Comparison of Weibull tail-coefficient estimators. REVSTAT Stat. J. 4, 163–188 (2006)
de Haan, L.: Fighting the arch-enemy with mathematics. Stat. Neerl. 44, 45–68 (1990)
de Haan, L., Ferreira, A.: Extreme value theory - An introduction. Springer (2006)
de Haan, L., Rootzén, H.: On the estimation of high quantiles. J. Stat. Plan. Inference 35(1), 1–13 (1993)
de Haan, L., Stadtmüller, U.: Generalized regular variation of second order. J. Austral. Math. Soc. (Ser. A) 61, 381–395 (1996)
Hall, P.: On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B 44(1), 37–42 (1982)
ISO: Petroleum and natural gas industries - Specific requirements for offshore structures Part 1: Metocean design and operating considerations. ISO/FDIS ISO/FDIS 19901–1:2005(E)
Klüppelberg, C.: On the asymptotic normality of parameter estimates for heavy Weibull-like tails. Preprint (1991)
Li, D., Peng, L., Yang, J.: Bias reduction for high quantiles. J. Stat. Plan. Inference 140(9), 2433–2441 (2010)
Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)
Smirnov, N.V.: Limit distributions for the terms of a variational series. Trudy Mat. Inst. Steklov. 25(1949) (1952). (Transl. Amer. Math. Soc. 11, 82-143)
Weissman, I.: Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc. 73, 812–815 (1978)
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de Valk, C. Approximation of high quantiles from intermediate quantiles. Extremes 19, 661–686 (2016). https://doi.org/10.1007/s10687-016-0255-3
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DOI: https://doi.org/10.1007/s10687-016-0255-3
Keywords
- Extreme value theory
- Quantile estimation
- High quantile
- Generalised Weibull tail limit
- log-GW tail limit
- Weibull tail limit
- Extended regular variation