On a class of M-estimators for Gaussian long-memory models

Abstract

We consider estimation for parametric stationary Gaussian models with long memory. The spectral density f(x; θ) is assumed to be characterised by a vector θ = (θ1, θ2, θ3,…, θm) such that θ2 = H ε (½, 1) and f(x; θ) is proportional to x1−2H as x tends to zero. An approximate maximum likelihood estimator based on the autoregressive representation of the process is proposed. Its asymptotic distribution is derived. More generally, the approach leads to a class of M-estimators for which a central limit theorem holds. By choosing an appropriate ψ-function, robustness against additive outliers can be achieved, while keeping high efficiency under the ideal model. This is illustrated by a small simulation study. A simple algorithm and an explicit formula for the efficiency, and thus for choosing an appropriate tuning parameter, are given for Hampel's redescending ψ-function.