Gaussian semiparametric estimation of long range dependence

Assuming the model f(A) GA1- 2H, as A -- 0 +, for the spectral densityo f a covariances tationaryp rocess,w e considera n estimateo f H E (0, 1) which maximizes an approximate form of frequency domain Gaussian likelihood, where discrete averaging is carried out over a neighbourhood of zero frequency which degenerates slowly to zero as sample size tends to infinityT. he estimate has several advantages. It is shown to be consistent under mild conditions. Under conditions which are not greatly stronger, it is shown to be asymptotically normal and more efficientt han previous estimates. Gaussianity is nowhere assumed in the asymptotict heory,t he limitingn ormal distributioni s of very simple form, involving a variance which is not dependent on unknown parameters, and the theory covers simultaneously the cases f(A) -x oc, f(A) -+ 0 and f(A) -+ C E (0, oc), as A -* 0. Monte Carlo evidence on finite-sample performance is reported, along with an application to a historical series of minimum levels of the River Nile.