DIFFERENTIABLE FUNCTIONALS AND SMOOTHED BOOTSTRAP

ANTONIO CUEVAS^{1} AND JUAN ROMO^{2}

1Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain

2Departamento de Estadística y Econometría, Universidad Carlos III de Madrid, 28903-Getafe (Madrid), Spain

(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 T_{n} = T (f̂_{n}) generated by a density functional T = T (f), f̂_{n} being a density estimator. As an application, a quick and easy proof of the asymptotic normality of ∫ f̂_{n}^{2}, (the plug-in estimator of the integrated squared density ∫ f^{2}) is given.

Key words and phrases: Smoothed bootstrap, differentiable statistical functionals, bootstrap validity, smoothed empirical process, integrated squared densities.