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.

翻译：我们研究了基于被高斯移动平均过程污染的高散观测的分数布朗模型的局部渐近性质。在统一框架下，我们同时考虑了低频率和高频率观测的情形，并证明了收敛速度 $n^{1/2} (\nu_n \Delta_n^{-H})^{-1/(2H+2K+1)}$ 是估计赫斯特指数 $H$ 的最优速率，其中 $\nu_n$ 为噪声强度，$\Delta_n$ 为采样频率，$K$ 为移动平均阶数。我们还推导了渐近有效方差，并构造了能达到该收敛速度和方差的估计量。这一理论分析得到了对估计方法全面数值分析的支撑，尤其说明了其在小样本情况下的有效性。