  We estimate the Hurst parameter $H \in (0,1)$ of a fractional Brownian motion
from discrete noisy data, observed along a high frequency sampling scheme. When
the intensity $\tau_n$ of the noise is smaller in order than $n^{-H}$ we
establish the LAN property with optimal rate $n^{-1/2}$. Otherwise, we
establish that the minimax rate of convergence is $(n/\tau_n^2)^{-1/(4H+2)}$
even when $\tau_n$ is of order 1. Our construction of an optimal procedure
relies on a Whittle type construction possibly pre-averaged, together with
techniques developed in Fukasawa et al. [Is volatility rough? arXiv:1905.04852,
2019]. We establish in all cases a central limit theorem with explicit
variance, extending the classical results of Gloter and Hoffmann [Estimation of
the Hurst parameter from discrete noisy data. The Annals of Statistics,
35(5):1947-1974, 2007].
