ABSTRACT
Long memory is an asymptotic property, namely the asymptotic decay of the correlations or the ultimate behavior of the spectral density as the frequency tends to zero. In the case of long memory, this condition does not hold. A new proof is needed to establish efficiency of the maximum likelihood estimation. Often one is interested in more than just the question of whether there is long memory in the data. In fact, in many applications, long memory is mainly a nuisance one has to deal with rather than the actual objective of the statistical analysis. Thus, P. Whittle's approximate maximum likelihood estimation (MLE) has the same asymptotic distribution as the exact MLE. It is therefore asymptotically efficient for Gaussian processes. It is well known that, under suitable regularity conditions, the MLE is asymptotically efficient in the sense of R. A. Fisher.
