There have been a number of papers written on semi-parametric estimation
methods of the long-memory exponent of a time series, some applied, others
theoretical. Some using Fourier methods, others using a wavelet-based
technique. In this paper, we compare the Fourier and wavelet approaches to the
local regression method and to the local Whittle method. We provide an overview
of these methods, describe what has been done, indicate the available results
and the conditions under which they hold. We discuss their relative strengths
and weaknesses both from a practical and a theoretical perspective. We also
include a simulation-based comparison. The software written to support this
work is available on demand and we illustrate its use at the end of the paper