## On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

##### Abstract

Approximate normality and unbiasedness of the maximum likelihood estimate (MLE) of the long-memory
parameter H of a fractional Brownian motion hold reasonably well for sample sizes as small as 20 if the
mean and scale parameter are known.We show in aMonte Carlo study that if the latter two parameters are
unknownthe bias and variance of theMLEofH both increase substantially.We also showthat the bias can be
reduced by using a parametric bootstrap procedure. In very large samples, maximum likelihood estimation
becomes problematic because of the large dimension of the covariance matrix that must be inverted. To
overcome this difficulty, we propose a maximum likelihood method based upon first differences of the
data. These first differences form a short-memory process. We split the data into a number of contiguous
blocks consisting of a relatively small number of observations. Computation of the likelihood function in
a block then presents no computational problem.We form a pseudo-likelihood function consisting of the
product of the likelihood functions in each of the blocks and provide a formula for the standard error of the
resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation
to the true standard error. The computation time required to obtain the estimate and its standard error from
large data sets is an order of magnitude less than that required to obtain the widely used Whittle estimator.
Application of the methodology is illustrated on two data sets

##### URI

http://hdl.handle.net/10394/16205http://dx.doi.org/10.1080/00949655.2012.732076

http://www.tandfonline.com/doi/abs/10.1080/00949655.2012.732076