  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.
