×

Strong convergence bounds of the Hill-type estimator under second-order regularly varying conditions. (English) Zbl 1095.62062

From the introduction: Suppose \(X_1,X_2,\dots\) are independent and identically distributed (iid) random variables with common distribution function (df) \(F\). Let \(M_n=\max\{X_1,\dots,X_n\}\) denote the maximum of the first \(n\) random variables and let \(w(F)=\sup\{x:F(x)<1 \}\) denote the upper end point of \(F\). The extreme value theory seeks norming constants \(a_n>0\), \(b_n\in{\mathfrak R}\), and a nondegenerate df \(G\) such that the df of a normalized version of \(M_n\) converges to \(G\), that is, \[ \text{Pr}((M_n-b_n)/a_n\leq x)=F^n(a_nx+b_n)\to G(x)\tag{1} \] as \(n\to\infty\). If this holds for suitable choices of \(a_n\) and \(b_n\) then it is said that \(G\) is an extreme value df and \(F\) is in the domain of attraction of \(G\), written as \(F\in D(G)\). For suitable constants \(a>0\) and \(b\in{\mathfrak R}\), one can write \[ G(ax+b)=G_\gamma(x)=\exp\bigl \{-(1+\gamma x)^{-1/\gamma} \bigr\}\tag{2} \] for all \(1+\gamma x> 0\) and \(\gamma\in{\mathfrak R}\). The distribution given by (2) is known as the extreme value distribution. Its practical applications have been wide-ranging: fire protection and insurance problems, models for extremely high temperatures, prediction of high return levels of wind speeds relevant for the design of civil engineering structures, models for extreme occurrences in Germany’s stock index, prediction of the behavior of solar proton peak fluxes, models for the failure strengths of load-sharing systems and window glasses, models for the magnitude of future earthquakes, analysis of corrosion failures of lead-sheathed cables at the Kennedy space center, prediction of the occurrence of geomagnetic storms, and estimation of the occurrence probability of giant freak waves in the sea area around Japan.
Each of the above problems requires estimation of the extremal index \(\gamma\) in (2). Several estimators for \(\gamma\) have been proposed in the extreme value theory literature. However, there has been little work on trying to study the convergence properties of the estimators for \(\gamma\). The question is: what is the penultimate form of the limit in (1)? Addressing this question is important because it will enable one to improve the modeling in each of the problems above. The aim of this paper is to consider the strong convergence rate of the Hill type estimator [B. M. Hill, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] for \(\gamma\) under the second-order regularly varying conditions.

MSC:

62G32 Statistics of extreme values; tail inference
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0323.62033
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] De Haan, L.; Galambos, J. (ed.), Extreme value statistics, 93-122 (1994), Massachusetts · doi:10.1007/978-1-4613-3638-9_6
[2] Deheuvels P: Strong laws for the[InlineEquation not available: see fulltext.]th order statistic when[InlineEquation not available: see fulltext.]. Probability Theory and Related Fields 1986,72(1):133-154. 10.1007/BF00343900 · Zbl 0592.60020 · doi:10.1007/BF00343900
[3] Deheuvels, P., Strong laws for the[InlineEquation not available: see fulltext.]th order statistic when[InlineEquation not available: see fulltext.]. II, No. 51, 21-35 (1989), New York · doi:10.1007/978-1-4612-3634-4_3
[4] Deheuvels P, Mason DM: The asymptotic behavior of sums of exponential extreme values.Bulletin des Sciences Mathematiques, Series 2 1988,112(2):211-233. · Zbl 0652.62016
[5] Dekkers ALM, de Haan L: On the estimation of the extreme-value index and large quantile estimation.The Annals of Statistics 1989,17(4):1795-1832. 10.1214/aos/1176347396 · Zbl 0699.62028 · doi:10.1214/aos/1176347396
[6] Dekkers ALM, Einmahl JHJ, de Haan L: A moment estimator for the index of an extreme-value distribution.The Annals of Statistics 1989,17(4):1833-1855. 10.1214/aos/1176347397 · Zbl 0701.62029 · doi:10.1214/aos/1176347397
[7] Drees H: On smooth statistical tail functionals.Scandinavian Journal of Statistics 1998,25(1):187-210. 10.1111/1467-9469.00097 · Zbl 0923.62032 · doi:10.1111/1467-9469.00097
[8] Hill BM: A simple general approach to inference about the tail of a distribution.The Annals of Statistics 1975,3(5):1163-1174. 10.1214/aos/1176343247 · Zbl 0323.62033 · doi:10.1214/aos/1176343247
[9] Pan J: Rate of strong convergence of Pickands’ estimator.Acta Scientiarum Naturalium Universitatis Pekinensis 1995,31(3):291-296. · Zbl 0834.62039
[10] Peng Z: An extension of a Pickands-type estimator.Acta Mathematica Sinica 1997,40(5):759-762. · Zbl 0908.60039
[11] Pickands J III: Statistical inference using extreme order statistics.The Annals of Statistics 1975,3(1):119-131. 10.1214/aos/1176343003 · Zbl 0312.62038 · doi:10.1214/aos/1176343003
[12] Resnick SI: Extreme Values, Regular Variation, and Point Processes, Applied Probability. A Series of the Applied Probability Trust. Volume 4. Springer, New York; 1987:xii+320.
[13] Wellner JA: Limit theorems for the ratio of the empirical distribution function to the true distribution function.Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1978,45(1):73-88. 10.1007/BF00635964 · Zbl 0382.60031 · doi:10.1007/BF00635964
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.