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

TL;DR

This paper compares the maximum likelihood and adapted absolute-moment methods for estimating the Hurst exponent in stationary fractional Brownian motion, highlighting their relative accuracy, computational efficiency, and practical utility.

Contribution

It introduces an adaptation of the absolute-moment method for stationary fractional Brownian motion and compares it to maximum likelihood estimation through simulations and real data.

Findings

Maximum likelihood is more accurate than the adapted absolute-moment method.

The adapted absolute-moment method is computationally more efficient.

The absolute-moment method helps visually confirm model specification.

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 $\theta$. 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.