Gaussian estimation of parametric spectral density with unknown pole

Abstract

We consider a parametric spectral density with power-law behavior about a fractional pole at the unknown frequency ω. The case of known ω, especially ω = 0, is standard in the long memory literature. When ω is unknown, asymptotic distribution theory for estimates of parameters, including the (long) memory parameter, is significantly harder. We study a form of Gaussian estimate. We establish n - consistency of the estimate of ω, and discuss its (non-standard) limiting distributional behavior. For the remaining parameter estimates, we establish √n-consistency and asymptotic normality.