International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1991 / Article

Open Access

Volume 14 |Article ID 710517 | 6 pages | https://doi.org/10.1155/S0161171291001060

Strong consistencies of the bootstrap moments

Received14 Nov 1990
Revised23 Jan 1991

Abstract

Let X be a real valued random variable with E|X|r+δ< for some positive integer r and real number, δ, 0<δr, and let {X,X1,X2,} be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost all wΩ, μr;n*(w)μr with probability 1. if limninfm(n)nβ>0 for some β>rδr+δ, where μr;n* is the bootstrap rth sample moment of the bootstrap sample some with sample size m(n) from the data set {X,X1,,Xn} and μr is the rth moment of X. The results obtained here not only improve on those of Athreya [3] but also the proof is more elementary.

Copyright © 1991 Hindawi Publishing Corporation. 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.


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