Estimation in autoregressive model with measurement error

Abstract

Consider an autoregressive model with measurement error: we observe $Z_i=X_i+{\epsilon }_i$, where $X_i$ is a stationary solution of the equation $X_i=f_{{\theta }^0}(X_{i-1})+{\xi }_i$. The regression function $f_{{\theta }^0}$ is known up to a finite dimensional parameter ${\theta }^0$. The distributions of $X_0$ and ${\xi }_1$ are unknown whereas the distribution of ${\epsilon }_1$ is completely known. We want to estimate the parameter ${\theta }^0$ by using the observations $Z_0,..,Z_n$. We propose an estimation procedure based on a modified least square criterion involving a weight function $w$, to be suitably chosen. We give upper bounds for the risk of the estimator, which depend on the smoothness of the errors density $f_{\epsilon }$ and on the smoothness properties of $w{f}_{\theta }$.