Estimation of Integral Functionals of a Density

Let be a smooth function of k + 2 variables. We shall investigate in this paper the rates of convergence of estimators of T (f) = (f (x), f' (x), , f^ (k) (x), x) dx when f belongs to some class of densities of smoothness s. We prove that, when s 2k + 14, one can define an estimator Tₙ of T (f), based on n i. i. d. observations of density f on the real line, which converges at the semiparametric rate 1/ n. On the other hand, when s < 2k + 14, T (f) cannot be estimated at a rate faster than n^- with = 4 (s - k) / 4s + 1. We shall also provide some extensions to the multidimensional case. Those results extend previous works of Levit, of Bickel and Ritov and of Donoho and Nussbaum on estimation of quadratic functionals.