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October, 1988 Rate of Convergence in Bootstrap Approximations
Peter Hall
Ann. Probab. 16(4): 1665-1684 (October, 1988). DOI: 10.1214/aop/1176991590


Let $X_1, \ldots, X_n$ be independent and identically distributed random variables with zero mean and unit variance. It is shown that the random bootstrap approximation to the distribution of $S \equiv n^{-1/2} \sum_jX_j$, converges to normality at precisely the same rate as $n^{-3/2}|\sum_jX^3_j| + n^{-2}\sum_jX^4_j$ converges to 0, up to terms of smaller order than $n^{-1/2}$. This result is used to explore properties of the bootstrap approximation under conditions weaker than existence of finite third moment. In most cases of that type it turns out that the bootstrap approximation to the distribution of $S$ is asymptotically equivalent to the normal approximation, so that the numerical expense of calculating the bootstrap approximation would not be justified. There also exist circumstances where the third moment is "almost" finite, yet the bootstrap approximation is asymptotically much worse than the simpler normal approximation. Necessary and sufficient conditions are given for a one-term Edgeworth expansion of the bootstrap approximation.


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Peter Hall. "Rate of Convergence in Bootstrap Approximations." Ann. Probab. 16 (4) 1665 - 1684, October, 1988.


Published: October, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0655.62015
MathSciNet: MR958209
Digital Object Identifier: 10.1214/aop/1176991590

Primary: 60F05
Secondary: 60G50 , 62G99

Keywords: Bootstrap approximation , central limit theorem , Edgeworth expansion , Normal approximation , rate of convergence

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 4 • October, 1988
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