Estimation in autoregressive models with surrogate data and validation data

Time-series data are often subject to measurement error, usually the result of needing to estimate the variable of interest. Generally, however, the relationship between the surrogate variables and the true variables can be rather complicated compared to the classical additive error structure usually assumed. In this article, we address the estimation of the parameters in autoregressive models in the presence of function measurement errors. We first develop a parameter estimation method with the help of validation data; this estimation method does not depend on functional form and the distribution of the measurement error. The proposed estimator is proved to be consistent. Moreover, the asymptotic representation and the asymptotic normality of the estimator are also derived, respectively. Simulation results indicate that the proposed method works well for practical situation.