Kybernetika 51 no. 6, 923-932, 2015

Bootstrap method for central and intermediate order statistics under power normalization

H. M. Barakat, E. M. Nigm and O. M. KhaledDOI: 10.14736/kyb-2015-6-0923

Abstract:

It has been known for a long time that for bootstrapping the distribution of the extremes under the traditional linear normalization of a sample consistently, the bootstrap sample size needs to be of smaller order than the original sample size. In this paper, we show that the same is true if we use the bootstrap for estimating a central, or an intermediate quantile under power normalization. A simulation study illustrates and corroborates theoretical results.

Keywords:

power normalization, weak consistency, central order statistics, bootstrap technique, intermediate order statistics

Classification:

62G32, 62F40

paper.pdf

References:

  1. K. B. Athreya and J. Fukuchi: Bootstrapping extremes of i.i.d. random variables. In: Conference on Extreme Value Theory and Application, Gaitherburg, Maryland 1993, Vol. 3, pp. 23-29.   CrossRef
  2. K. B. Athreya and J. Fukuchi: Confidence interval for end point of a c.d.f, via bootstrap. J. Statist. Plann. Inf. 58 (1997), 299-320.   DOI:10.1016/s0378-3758(96)00087-0
  3. H. M. Barakat and M. A. El-Shandidy: On general asymptotic behaviour of order statistics with random index. Bull. Malays. Math. Sci. Soc. 27 (2004), 169-183.   CrossRef
  4. H. M. Barakat and A. R. Omar: Limit theorems for order statistics under nonlinear normalization. J. Statist. Plann. Inf. 141 (2011), 524-535.   DOI:10.1016/j.jspi.2010.06.031
  5. H. M. Barakat and A. R. Omar: On limit distributions for intermediate order statistics under power normalization. Math. Methods Statist. 20 (2011), 365-377.   DOI:10.3103/s1066530711040053
  6. H. M. Barakat, E. M. Nigm and M. E. El-Adll: Comparison between the rates of convergence of extremes under linear and under power normalization. Statist. Papers 51 (2010), 149-164.   DOI:10.1007/s00362-008-0128-1
  7. H. M. Barakat, E. M. Nigm and O. M. Khaled: Extreme value modeling under power normalization. Applied Math. Modelling 37 (2013), 10162-10169.   DOI:10.1016/j.apm.2013.05.045
  8. H. M. Barakat, E. M. Nigm, O. M. Khaled and F. A. Momenkhan: Bootstrap order statistics and modeling study of the air pollution. Commun. Statist. Simul. Comput. 44 (2015), 1477-1491.   DOI:10.1080/03610918.2013.805051
  9. D. M. Chibisov: On limit distributions of order statistics. Theory Probab. Appl. 9 (1964), 142-148.   DOI:10.1137/1109021
  10. B. Efron: Bootstrap methods: Another look at the Jackknife. Ann. Statist. 7 (1979), 1-26.   DOI:10.1214/aos/1176344552
  11. N. R. Mohan and S. Ravi: Max domains of attraction of univariate and multivariate $p$-max stable laws. Theory Probab. Appl. 37 (1992), 632-643.   DOI:10.1137/1137119
  12. E. M. Nigm: Bootstrapping extremes of random variables under power normalization. Test 15 (2006), 257-269.   DOI:10.1007/bf02595427
  13. E. Pancheva: Limit theorems for extreme order statistics under nonlinear normalization. Lecture Notes in Math. 1155 (1984), 284-309.   DOI:10.1007/bfb0074824
  14. N. V. Smirnov: Limit distributions for the terms of a variational series. Amer. Math. Soc. Transl. Ser. 11 (1952), 82-143.   CrossRef