On Estimation of Hurst Parameter Under Noisy Observations
针对含噪声的离散采样分数阶积分过程，提出用预平均法降噪、滤波法去除强依赖，再用平滑数据估计赫斯特参数，并证明了估计量的一致性和渐近正态性。
It is widely accepted that some financial data exhibit long memory or long dependence, and that the observed data usually possess noise. In the continuous time situation, the factional Brownian motion <i>B<sup>H</sup></i> and its extension are an important class of models to characterize the long memory or short memory of data, and Hurst parameter <i>H</i> is an index to describe the degree of dependence. In this article, we estimate the Hurst parameter of a discretely sampled fractional integral process corrupted by noise. We use the preaverage method to diminish the impact of noise, employ the filter method to exclude the strong dependence, and obtain the smoothed data, and estimate the Hurst parameter by the smoothed data. The asymptotic properties such as consistency and asymptotic normality of the estimator are established. Simulations for evaluating the performance of the estimator are conducted. Supplementary materials for this article are available online.