(Received May 15, 1995; revised May 7, 1996)
Abstract. 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 \int ^fn2 , (the plug-in estimator of the integrated squared density \int f 2 ) is given.
Key words and phrases: Smoothed bootstrap, differentiable statistical functionals, bootstrap validity, smoothed empirical process, integrated squared densities.