On a non-parametric analogue of the information matrix

For a class of differentiable functions $\Phi(F)$ of distributions $F$, an analogue of the information matrix $I(F)$ is considered. In terms of matrix $I(F)$, bounds for risks in estimating $\Phi(F)$ are obtained; this is an extension, to the non-parametric case, of a result of J. Hajek [2]. Some examples are discussed including estimation of $\Phi(F)$ under the restriction that the values of differentiable functions $\Psi(F)$ are known.