On Parameter Estimation from Indirect Observations

We consider the problem of parameter estimation in the situation where the observation $Y_\varepsilon(t)$ is the sum of a known function $Phi$ of an unobservable process $Y_\varepsilon(t)$ and the Gaussian white noise of small intensity $\varepsilon$. The process $Y_\varepsilon(t)$ is also the sum of a signal depending on an unknown parameter $\theta\in R^d$ and the Gaussian process with small correlation operator. The property of local asymptotic normality (LAN), lower and upper bounds for the estimation quality, and the construction of an asymptotically efficient estimate are established for this model.