ESTIMATION OF INTEGRAL FUNCTIONALS OF A DENSITY

Let phi be a smooth function of k + 2 variables. We shall investigate in this paper the rates of convergence of estimators of T(f) = integral phi(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 greater than or equal to 2k + 1/4, one can define an estimator (T) over cap(n) of T(f), based on n i.i.d. observations of density f on the real line, which converges at the semiparametric rate 1/root n. On the other hand, when s < 2k + 1/4, T(f) cannot be estimated at a rate faster than n(-gamma) with gamma = 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.