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Article Dans Une Revue ESAIM: Probability and Statistics Année : 2014

## Estimation in autoregressive model with measurement error

Jérôme Dedecker
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Marie-Luce Taupin
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#### Résumé

Consider an autoregressive model with measurement error: we observe $Z_i=X_i+\varepsilon_i$, 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 $\varepsilon_1$ is completely known. We want to estimate the parameter $\theta^0$ by using the observations $Z_0,\ldots,Z_n$. 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 $f_\varepsilon$ and on the smoothness properties of $w f_\theta$.

### Dates et versions

hal-00591114 , version 1 (06-05-2011)
hal-00591114 , version 2 (24-10-2011)

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• HAL Id : hal-00591114 , version 2
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### Citer

Jérôme Dedecker, Adeline Samson, Marie-Luce Taupin. Estimation in autoregressive model with measurement error. ESAIM: Probability and Statistics, 2014, ESAIM Probability and Statistics 18, pp.277-307. ⟨hal-00591114v2⟩

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