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

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 $\theta$. We present an adaptation of the absolute-moment method to this framework and we compare it to the maximum likelihood method, with simulations. The conclusion is mainly in favour of the adapted absolute-moment method for several reasons: it makes it possible to confirm visually that the model is well specified, it is computationally more performing, the estimation of $\theta$ is more accurate.