On estimation of functionals of the probability density function and its derivatives

For functionals of the type $I=\int_{-\infty}^\infty H(f(y)),f'(y),\dots,f^{(r)}(y))\,dy$ the estimates $I_N=\int_{-k_N}^{k_N}H(f_N(y),\dots,f_N^{(r)}(y))\,dy$ are considered. Here $f_N(y),\dots,f_N^{(r)}(y)$ are nonparametric estimates of the density and of its derivatives introduced by Rosenblatt and studied by Parzen, Bhattacharya, Nadaraya and others. Theorems on convergence of the estimates with probability one are proved for Fisher's information, the entropy and the integral of the squared density. Convergence in probability are also investigated.