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].
翻译:根据高频采样方案观测到的离散噪音数据,我们估算出一个微小的棕色运动的赫斯特参数$H(0,1美元)美元。当噪音的强度小于0.00美元时,我们用最佳的速率建立局域网属性,最高速率为$N ⁇ -1/2美元。否则,我们确定微小的汇合率是$(n/tau_n)2)、 ⁇ -1/(4H+2)美元,即使美元是按顺序排列的1美元。我们建造一个最佳程序依靠惠特尔(Whittle)型的建筑,可能为平均前,加上Fukasawa等人(Fukasawa等人(Fukasawa等人)开发的技术)[易挥发性:axiv:1905.4852,20199]。我们在所有情况下都设定一个有明显差异的中心界限,扩大Gloter和Hoffmann的典型结果[根据离散噪音数据对赫斯特参数的刺激。Annals统计,35(5):1947-1974,2007]。