A note on proving that the (modified) bootstrap works. (English) Zbl 0623.62041

Swanepoel, Jan W. H.

Commun. Stat., Theory Methods 15, 3193-3203 (1986).

Let \(X_ 1,...,X_ n\) be i.i.d. random variables (rv’s) and let \(X^*_ 1,...,X^*_ m\) be a bootstrap sample of i.i.d. rv’s with distribution function (df) \(F_ n\), where \(F_ n\) is the empirical df based on \(X_ 1,...,X_ n\). Specifying \(m=m(n)\) as some suitable function of n it is shown first that the bootstrap also works for the counter-examples given by P. J. Bickel and D. Freedman, Ann. Stat. 9, 1196-1217 (1981; Zbl 0449.62034).
Secondly, based on a result of W. Stute, Ann. Probab. 10, 86-107 (1982; Zbl 0489.60038) on the oscillation behavior of empirical processes a method is suggested which can be used to show that the bootstrap method of distribution approximation is asymptotically valid.

Reviewer: P.Gaenssler

MSC:

62G30	Order statistics; empirical distribution functions
62E20	Asymptotic distribution theory in statistics
62D05	Sampling theory, sample surveys
65C05	Monte Carlo methods
60F17	Functional limit theorems; invariance principles
62F25	Parametric tolerance and confidence regions

Keywords:

quantile process; empirical distribution; bootstrap; empirical processes; approximation

References:

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