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On the bootstrap and confidence intervals. (English) Zbl 0611.62047
It is known that the bootstrap can be viewed as an empiric one term Edgeworth inversion. The present paper shows that such an inversion given by bootstrap is smooth and does not require preliminary theoretical calculations of the Edgeworth expansion. As such the expansions obtained via bootstrap are useful in describing coverage probabilities or significance levels of a wide class of ”Studentized” statistics. It is shown that the bootstrap may be iterated to give approximation to any desired degree of accuracy. This result is of theoretical nature and explains the encouraging performance of bootstrap methods. Bootstrap iteration involves simulations of simulations and is extremely labourious to implement. However in another paper [see the following review, Zbl 0611.62048] the author provides useful information about the number of simulations required.
Reviewer: B.K.Kale

MSC:
62G15Nonparametric tolerance and confidence regions
62E20Asymptotic distribution theory in statistics
62G05Nonparametric estimation
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
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