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Estimation in autoregressive model with measurement error

J. Dedecker,A. Samson,M. Taupin

Published 2011 in arXiv: Statistics Theory

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 $w f_\theta$.

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