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  • Research Article
  • Open Access

Strong convergence bounds of the Hill-type estimator under second-order regularly varying conditions

Journal of Inequalities and Applications20062006:95124

  • Received: 22 April 2005
  • Accepted: 10 July 2005
  • Published:


Bounds on strong convergences of the Hill-type estimator are established under second-order regularly varying conditions.


  • Strong Convergence
  • Convergence Bound


Authors’ Affiliations

Department of Mathematics, Southwest Normal University, Chongqing, 400715, China
Department of Statistics, University of Nebraska–Lincoln, Lincoln, NE 68583, USA


  1. De Haan L: Extreme value statistics. In Extreme Value Theory and Applications. Edited by: Galambos J. Kluwer Academic, Massachusetts; 1994:93–122.View ArticleGoogle Scholar
  2. Deheuvels P: Strong laws for the th order statistic when . Probability Theory and Related Fields 1986,72(1):133–154. 10.1007/BF00343900MathSciNetView ArticleMATHGoogle Scholar
  3. Deheuvels P: Strong laws for the th order statistic when . II. In Extreme Value Theory (Oberwolfach, 1987), Lecture Notes in Statistics. Volume 51. Springer, New York; 1989:21–35.View ArticleGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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/1176347396MathSciNetView ArticleMATHGoogle Scholar
  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/1176347397MathSciNetView ArticleMATHGoogle Scholar
  7. Drees H: On smooth statistical tail functionals. Scandinavian Journal of Statistics 1998,25(1):187–210. 10.1111/1467-9469.00097MathSciNetView ArticleMATHGoogle Scholar
  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/1176343247MathSciNetView ArticleMATHGoogle Scholar
  9. Pan J: Rate of strong convergence of Pickands' estimator. Acta Scientiarum Naturalium Universitatis Pekinensis 1995,31(3):291–296.MathSciNetMATHGoogle Scholar
  10. Peng Z: An extension of a Pickands-type estimator. Acta Mathematica Sinica 1997,40(5):759–762.MathSciNetMATHGoogle Scholar
  11. Pickands J III: Statistical inference using extreme order statistics. The Annals of Statistics 1975,3(1):119–131. 10.1214/aos/1176343003MathSciNetView ArticleMATHGoogle Scholar
  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.Google Scholar
  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/BF00635964MathSciNetView ArticleMATHGoogle Scholar


© Z. Peng and S. Nadarajah. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.