Strong law for the bootstrap

Athreya, K. B. (1983) Strong law for the bootstrap Statistics & Probability Letters, 1 (3). pp. 147-150. ISSN 0167-7152

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Let X1, X2, X3, ... be i.i.d. r.v. with E|X1| < ∞, E X1 = μ. Given a realization X = (X1,X2,...) and integers n and m, construct Yn,i, I = 1, 2, ..., m as i.i.d. r.v. with conditional distribution P∗(Yn,i = Xj) = 1/n for 1≤ j ≤ n. (P∗ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X1 are given to ensure the almost sure convergence of (1/m)∑mi=1 Yn,i to μ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Bootstrap; Strong Law
ID Code:1117
Deposited On:05 Oct 2010 12:54
Last Modified:12 May 2011 10:15

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