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Let $(X_1,\ldots ,X_n)$ be an i.i.d. sequence of random variables in $\R^d$,$d\geq 1$, for some function $\varphi:\R^d\r \R$, under regularity conditions,we show that \begin{align*} n^{1/2} \left(n^{-1} \sum_{i=1}^n\frac{\varphi(X_i)}{\w f^{(i)}(X_i)}-\int_{} \varphi(x)dx \right)\overset{\P}{\lr} 0, \end{align*} where $\w f^{(i)}$ is the classicalleave-one-out kernel estimator of the density of $X_1$. This result is strikingbecause it speeds up traditional rates, in root $n$, derived from the centrallimit theorem when $\w f^{(i)}=f$. As a consequence, it improves the classicalMonte Carlo procedure for integral approximation. The paper mainly addressedwith theoretical issues related to the later result (rates of convergence,bandwidth choice, regularity of $\varphi$) but also interests some statisticalapplications dealing with random design regression. In particular, we providethe asymptotic normality of the estimation of the linear functionals of aregression function on which the only requirement is the H\"older regularity.This leads us to a new version of the \textit{average derivative estimator}introduced by H\"ardle and Stoker in \cite{hardle1989} which allows for\textit{dimension reduction} by estimating the \textit{index space} of aregression.

On the acceleration of some empirical means with application to nonparametric regression