Differentiable functionals and smoothed bootstrap.

The differentiability properties of statistical functionals have several interesting applications. We are concerned with two of them. First, we prove a result on asymptotic validity for the so-called smoothed bootstrap (where the artificial samples are drawn from a density estimator instead of being resampled from the original data). Our result can be considered as a smoothed analog of that obtained by Parr (1985, Stat. Probab. Lett., 3, 97-100) for the standard, unsmoothed bootstrap. Second, we establish a result on asymptotic normality for estimators of type Tn=T(fn) generated by a density functional T=T(f) fn being a density estimator. As an application, a quick and easy proof of the asymptotic normality of fn2 , (the plug-in estimator of the integrated squared density f2 ) is given.Grateful to Ricardo Fraiman and Ricardo Cao for helpful comments concerning technical details in the proofs. We also gratefully acknowledge the careful report of an anonymous reference. This work was partially supported by Spanish grants PB94-0179 (A. Cuevas) and PB93-0232(J. Romo