For a multidimensional It\^o semimartingale, we consider the problem of estimating integrated volatility functionals. Jacod and Rosenbaum (2013) studied a plug-in type of estimator based on a Riemann sum approximation of the integrated functional and a spot volatility estimator with a forward uniform kernel. Motivated by recent results that show that spot volatility estimators with general two-side kernels of unbounded support are more accurate, in this paper, an estimator using a general kernel spot volatility estimator as the plug-in is considered. A biased central limit theorem for estimating the integrated functional is established with an optimal convergence rate. Unbiased central limit theorems for estimators with proper de-biasing terms are also obtained both at the optimal convergence regime for the bandwidth and when applying undersmoothing. Our results show that one can significantly reduce the estimator's bias by adopting a general kernel instead of the standard uniform kernel. Our proposed bias-corrected estimators are found to maintain remarkable robustness against bandwidth selection in a variety of sampling frequencies and functions.

翻译：针对多维 Itô 半鞅过程，本文研究积分波动泛函的估计问题。Jacod 与 Rosenbaum (2013) 提出了一种基于黎曼和近似积分泛函的插件型估计量，其现货波动率估计量采用前向均匀核。受近期研究启发——该研究表明采用无界支撑的一般双侧核的现货波动率估计量具有更高精度——本文提出一种以一般核现货波动率估计量作为插件的估计量。我们建立了估计积分泛函的带有偏差的中心极限定理，并获得了最优收敛速率。同时，在带宽处于最优收敛状态及采用欠平滑策略时，通过引入适当的去偏项，我们也得到了无偏的中心极限定理。研究结果表明，采用一般核替代标准均匀核可显著降低估计量的偏差。本文提出的偏差校正估计量在不同采样频率与函数形式下，对带宽选择均表现出显著的稳健性。