Estimation in autoregressive model with measurement error

¹ Laboratoire MAP5 UMR CNRS 8145, Université Paris Descartes, Sorbonne Paris Cité, Paris cedex 6, France
² Laboratoire Statistique et Génome, UMR CNRS 8071-USC INRA, Université d’Évry Val d’Essonne, Évry, France
marie-luce.taupin@genopole.cnrs.fr

Received: 24 October 2011
Revised: 14 February 2013

Abstract

Consider an autoregressive model with measurement error: we observe Z_i = X_i + ε_i, where the unobserved X_i is a stationary solution of the autoregressive equation X_i = g_θ⁰(X_{i − 1}) + ξ_i. The regression function g_θ⁰ is known up to a finite dimensional parameter θ⁰ to be estimated. The distributions of ξ₁ and X₀ are unknown and g_θ belongs to a large class of parametric regression functions. The distribution of ε₀ 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 of θ⁰, for a large class of regression functions and various noise distributions.