ESTIMATION OF THE SECOND ORDER PARAMETER CHARACTERIZING THE TAIL BEHAVIOR OF PROBABILITY DISTRIBUTIONS: ASYMPTOTIC NORMALITY

  • Yun Seok-Hoon (Department of Applied Statistics, University of Suwon)
  • Published : 2006.03.01

Abstract

Yun (2005) introduced an estimator of the second order parameter characterizing the tail behavior of probability distributions and proved its consistency. In this paper we prove its asymptotic normality under a third order condition.

Keywords

References

  1. DE HAAN, L. (1984). 'Slow variation and characterization of domains of attraction' In Statistical Extremes and Applications (J. Tiago de Oliveira, ed.), 31-48, Reidel, Dordrecht
  2. DE HAAN, L. AND STADTMULLER, U. (1996). 'Generalized regular variation of second order', Journal of the Australian Mathematical Society. Ser. A, 61, 381-395 https://doi.org/10.1017/S144678870000046X
  3. DEKKERS, A. L. M. AND DE HAAN, L. (1989). 'On the estimation of the extreme-value index and large quantile estimation', The Annals of Statistics, 17, 1795-1832 https://doi.org/10.1214/aos/1176347396
  4. DEKKERS, A. L. M., EINMAHL, J. H. J. AND DE HAAN, L. (1989). 'A moment estimator for the index of an extreme-value distribution', The Annals of Statistics, 17, 1833-1855 https://doi.org/10.1214/aos/1176347397
  5. DRAISMA, G., DE HAAN, L., PENG, L. AND PEREIRA, T. T. (1999). 'A bootstrap-based method to achieve optimality in estimating the extreme-value index', Extremes, Engineering and Economics, 2, 367-404 https://doi.org/10.1023/A:1009900215680
  6. DREES, H. (1995). 'Refined Pickands estimators of the extreme value index', The Annals of Statistics, 23, 2059-2080 https://doi.org/10.1214/aos/1034713647
  7. GOMES, M. I., DE HAAN, L. AND PENG, L. (2002). 'Semi-parametric estimation of the second order parameter in statistics of extremes', Extremes, 5, 387-414 https://doi.org/10.1023/A:1025128326588
  8. HILL, B. M. (1975). 'A simple general approach to inference about the tail of a distribution', The Annals of Statistics, 3, 1163-1174 https://doi.org/10.1214/aos/1176343247
  9. PICKANDS, J. (1975). 'Statistical inference using extreme order statistics', The Annals of Statistics, 3, 119-131 https://doi.org/10.1214/aos/1176343003
  10. YUN, S. (2002). 'On a generalized Pickands estimator of the extreme value index', Journal of Statistical Planning and Inference, 102, 389-409 https://doi.org/10.1016/S0378-3758(01)00100-8
  11. YUN, S. (2005). 'Estimation of the second order parameter characterizing the tail behavior of probability distributions: Consistency', Journal of the Korean Statistical Society, 34, 273-280