Estimation of Integral Functionals of a Density

This is the authors' abstract. We don't add key points for this paper.

Let φ be a smooth function of $k + 2$ variables. We shall investigate in this paper the rates of convergence of estimators of T(f) = ∫φ(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 + 1/4, one can define an estimator T̂ₙ 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 + 1/4, T(f) cannot be estimated at a rate faster than n-γ with γ = 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.