A comparison of maximum likelihood and absolute moments for the estimation of Hurst exponents in a stationary framework

Abstract

The absolute-moment method is widespread for estimating the Hurst exponent of a fractional Brownian motion X. But this method is biased when applied to a stationary version of X, in particular an inverse Lamperti transform of X, with a linear time contraction of parameter θ. We present an adaptation of the absolute-moment method to this framework and we compare it to the maximum likelihood method, with simulations and an application to a financial time series. While it appears that the maximum-likelihood method is more accurate than the adapted absolute-moment estimation, this last method is not uninteresting for two reasons: it makes it possible to confirm visually that the model is well specified and it is computationally more performing.