Moment estimator for an AR(1) model driven by a long memory Gaussian noise

Abstract

In this paper, we consider an inference problem for the first order autoregressive process driven by a given long memory stationary Gaussian noise. Suppose that the covariance function of the noise is positive and can be expressed as |k|2H−2 times a positive function slowly varying at infinity. The fractional Gaussian noise and the fractional ARIMA(0, d, 0) model with d∈(0,12) and some others Gaussian noise are special examples that satisfy this assumption. We propose a moment estimator and prove the strong consistency, the asymptotic normality and the almost sure central limit theorem. Moreover, we give the upper Berry–Esséen bound by means of Fourth moment theorem.