Estimation of integral functionals of a density and its derivatives

Abstract

We consider the problem of estimating a functional of a density of the type $\int \phi (f,f^{\prime },...,f^{\left(k\right)},.)$. The estimation of $\int \phi (f,.)$ has already been studied by the author: starting from efficient estimators of linear and quadratic functionals of $f$ and its derivatives and using a Taylor expansion of $\phi$, we construct estimators which achieve the $n^{-1/2}$ rate whenever $f$ is smooth enough. Moreover, we show that these estimators are efficient. We also obtain the optimal rate of convergence when the $n^{-1/2}$ rate is not achievable and when $k>0$. Concerning the estimation of quadratic functionals, more precisely of integrated squared density derivatives, Bickel and Ritov have already constructed efficient estimators. Here we propose an alternative construction based on projections, an approach which seems more natural.