Previous |  Up |  Next


bootstrap technique; power normalization; weak consistency; central order statistics; intermediate order statistics
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.
[1] Athreya, K. B., Fukuchi, J.: Bootstrapping extremes of i.i.d. random variables. In: Conference on Extreme Value Theory and Application, Gaitherburg, Maryland 1993, Vol. 3, pp. 23-29.
[2] Athreya, K. B., Fukuchi, J.: 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 | MR 1450018
[3] Barakat, H. M., El-Shandidy, M. A.: On general asymptotic behaviour of order statistics with random index. Bull. Malays. Math. Sci. Soc. 27 (2004), 169-183. MR 2124771 | Zbl 1185.60020
[4] Barakat, H. M., Omar, A. R.: Limit theorems for order statistics under nonlinear normalization. J. Statist. Plann. Inf. 141 (2011), 524-535. DOI 10.1016/j.jspi.2010.06.031 | MR 2719515
[5] Barakat, H. M., Omar, A. R.: On limit distributions for intermediate order statistics under power normalization. Math. Methods Statist. 20 (2011), 365-377. DOI 10.3103/s1066530711040053 | MR 2886642
[6] Barakat, H. M., Nigm, E. M., El-Adll, M. E.: 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 | MR 2556592 | Zbl 1247.60030
[7] Barakat, H. M., Nigm, E. M., Khaled, O. M.: Extreme value modeling under power normalization. Applied Math. Modelling 37 (2013), 10162-10169. DOI 10.1016/j.apm.2013.05.045 | MR 3125527
[8] Barakat, H. M., Nigm, E. M., Khaled, O. M., Momenkhan, F. A.: Bootstrap order statistics and modeling study of the air pollution. Commun. Statist. Simul. Comput. 44 (2015), 1477-1491. DOI 10.1080/03610918.2013.805051 | MR 3290573
[9] Chibisov, D. M.: On limit distributions of order statistics. Theory Probab. Appl. 9 (1964), 142-148. DOI 10.1137/1109021 | MR 0165633
[10] Efron, B.: Bootstrap methods: Another look at the Jackknife. Ann. Statist. 7 (1979), 1-26. DOI 10.1214/aos/1176344552 | MR 0515681 | Zbl 0406.62024
[11] Mohan, N. R., Ravi, S.: Max domains of attraction of univariate and multivariate $p$-max stable laws. Theory Probab. Appl. 37 (1992), 632-643. DOI 10.1137/1137119 | MR 1210055 | Zbl 0774.60029
[12] Nigm, E. M.: Bootstrapping extremes of random variables under power normalization. Test 15 (2006), 257-269. DOI 10.1007/bf02595427 | MR 2278954 | Zbl 1131.62039
[13] Pancheva, E.: Limit theorems for extreme order statistics under nonlinear normalization. Lecture Notes in Math. 1155 (1984), 284-309. DOI 10.1007/bfb0074824 | MR 0825331
[14] Smirnov, N. V.: Limit distributions for the terms of a variational series. Amer. Math. Soc. Transl. Ser. 11 (1952), 82-143. MR 0047277
Partner of
EuDML logo