Estimation in autoregressive model with measurement error

Dedecker, Jérôme; Samson, Adeline; Taupin, Marie-Luce

ESAIM: Probability and Statistics, Tome 18 (2014), pp. 277-307.

Consider an autoregressive model with measurement error: we observe $Z_i=X_i+{\epsilon }_i$, where the unobserved $X_i$ is a stationary solution of the autoregressive equation $X_i=g_{{\theta }^0}\left(X_{i-1}\right)+{\xi }_i$. The regression function $g_{{\theta }^0}$ is known up to a finite dimensional parameter ${\theta }^0$ to be estimated. The distributions of ${\xi }_1$and $X_0$ are unknown and $g_{\theta }$ belongs to a large class of parametric regression functions. The distribution of ${\epsilon }_0$ is completely known. We propose an estimation procedure with a new criterion computed as the Fourier transform of a weighted least square contrast. This procedure provides an asymptotically normal estimator $\widehat{\theta }$ of ${\theta }^0$, for a large class of regression functions and various noise distributions.

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