Estimation in autoregressive model with measurement error

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

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

Résumé

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.

Détail

DOI : 10.1051/ps/2013037

Classification : 62J02, 62F12, 62G05, 62G20
Keywords: autoregressive model, Markov chain, mixing, deconvolution, semi-parametric model