Asymptotic Efficiency for Fractional Brownian Motion with general noise

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 $n^{1\mathrm{/}2}({\nu }_n{\mathrm{\Delta }}_n^{-H}{)}^{-1\mathrm{/}(2H+2K+1)}$ is optimal for estimating the Hurst index $H$, where ${\nu }_n$ is the noise intensity, ${\mathrm{\Delta }}_n$ 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.