On-line Estimation of Smooth Signals with Partial Observation

The paper concerns the estimation of a smooth signal $S(t)$ and its derivatives in the presence of a noise depending on a small parameter $\varepsilon$ based on a partial observation. A nonlinear Kalman-type filter is proposed to perform on-line estimation. For the signal $S$ in a given class of smooth functions, the convergence rate for the estimation risks, as $\varepsilon\to0$, is obtained. It is proved that such rates are optimal in a minimax sense. In contrast to the complete observation case, the rates are reduced, due to incomplete information.