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On the number of bootstrap simulations required to construct a confidence interval. (English) Zbl 0611.62048
Suppose we obtain one-sided confidence intervals for a parameter $$\theta$$ by ”smooth” Studentized statistic using B bootstrap simulations. It is shown that the reduction of error of coverage probability from $$O(n^{- 1/2})$$ to $$O(n^{-1})$$ is available uniformly in B provided nominal coverage probability is a multiple of $$(B+1)^{-1}$$. This improvement is possible even when B is fixed and n is allowed to increase. Further it is shown that for large n, the simulated statistic values behave like random observations from a continuous distribution unless B increases faster than any power of n. The effect of discrete nature of the bootstrap statistic is felt only if B increases exponentially with n.
Reviewer: B.K.Kale

MSC:
 62G15 Nonparametric tolerance and confidence regions 62E20 Asymptotic distribution theory in statistics 65C99 Probabilistic methods, stochastic differential equations
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