Hall, Peter On the bootstrap and confidence intervals. (English) Zbl 0611.62047 Ann. Stat. 14, 1431-1452 (1986). 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 Cited in 2 ReviewsCited in 83 Documents MSC: 62G15 Nonparametric tolerance and confidence regions 62E20 Asymptotic distribution theory in statistics 62G05 Nonparametric estimation 65C99 Probabilistic methods, stochastic differential equations Keywords:confidence intervals; sample mean; sample correlation coefficient; maximum likelihood estimators expressible as functions of vector means; error of arbitrarily small order; central limit theorem; rates of convergence; order of approximation; Studentized statistics; bootstrap; Edgeworth inversion; Edgeworth expansion; coverage probabilities; significance levels; Bootstrap iteration Citations:Zbl 0611.62048 × Cite Format Result Cite Review PDF Full Text: DOI