Nonparametric Estimation of Functionals of the Derivatives of a Signal Observed in White Gaussian Noise

We consider the estimation of an integral functional of the signal $S$ and its derivatives up to order $p_m$ when the observable quantity is the result of transmission of $S$ through a communication channel with white Gaussian noise of low intensity $\varepsilon^2$. Nonparametric estimators of $S$ and $S^{(k)}$ in this case are known to have variance $\Delta^2\gg\varepsilon^2$. Yet a differentiable functional $F$ often may be estimated asymptotically efficiently with $\Delta^2\asymp\varepsilon^2$. We obtain nearly necessary conditions on the a priori known smoothness of the signal $\beta$, the smoothness of the derivative of the functional $\gamma$ and $p_m$ that ensure asymptotically (for $\varepsilon\to 0$) efficient estimation of $F$. The form of this estimator is given.