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DIFFERENTIABLE FUNCTIONALS AND

SMOOTHED BOOTSTRAP

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

^{1} *Departamento de Matemáticas,
Universidad Autónoma de Madrid, 28049-Madrid, Spain*

^{2} *Departamento 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*(^{^}*f*_{n}) generated by a density functional
*T* = *T*(*f*) , ^{^}*f*_{n} being a density estimator. As an
application, a quick and easy proof of the asymptotic
normality of \int ^{^}*f*_{n}^{2} , (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.

**Source**
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