  We investigate the Local Asymptotic Property for fractional Brownian models
based on discrete observations contaminated by a Gaussian moving average
process. We consider both situations of low and high-frequency observations in
a unified setup and we show that the convergence rate $n^{1/2} (\nu_n
\Delta_n^{-H})^{-1/(2H+2K+1)}$ is optimal for estimating the Hurst index $H$,
where $\nu_n$ is the noise intensity, $\Delta_n$ is the sampling frequency and
$K$ is the moving average order. We also derive asymptotically efficient
variances and we build an estimator achieving this convergence rate and
variance. This theoretical analysis is backed up by a comprehensive numerical
analysis of the estimation procedure that illustrates in particular its
effectiveness for finite samples.
