Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series

Abstract

A strongly dependent Gaussian sequence has a spectral density f(x,θ)f(x, \theta) satisfying f(x,θ)∼|x|−α(θ)Lθ(x)f(x, \theta) \sim |x|^{-\alpha(\theta)} L_\theta(x) as x→0x \rightarrow 0, where 0 <10 < \alpha(\theta) < 1 and Lθ(x)L_\theta(x) varies slowly at 0. Here θ\theta is a vector of unknown parameters. An estimator for θ\theta is proposed and shown to be consistent and asymptotically normal under appropriate conditions. These conditions are satisfied by fractional Gaussian noise and fractional ARMA, two examples of strongly dependent sequences.