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 generated by a density functional being a density estimator. As an application, a quick and easy proof of the asymptotic normality of , (the plug-in estimator of the integrated squared density ) is given.