Estimation in autoregressive model with measurement error

TL;DR

This paper develops a new estimation method for autoregressive models with measurement error, leveraging a modified least squares approach and analyzing its risk bounds based on error and weight function smoothness.

Contribution

It introduces a novel estimation procedure for autoregressive models with measurement error, accounting for unknown distributions and providing risk bounds.

Findings

The proposed estimator achieves risk bounds depending on error density smoothness.

The method effectively handles unknown distributions of the process and noise.

Upper bounds on estimation risk are derived based on smoothness assumptions.

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 $w f_\theta$.