Rate of Convergence in Bootstrap Approximations

Abstract

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.