ESTIMATION OF INTEGRATED VOLATILITY FUNCTIONALS WITH KERNEL SPOT VOLATILITY ESTIMATORS
针对多维伊藤半鞅过程，提出使用一般核函数的瞬时波动率估计量来估计积分波动率泛函，相比标准均匀核能显著降低偏差，并建立了有偏和无偏估计量的中心极限定理。
For a multidimensional Itô semimartingale, we consider the problem of estimating integrated volatility functionals. Jacod and Rosenbaum (2013, The Annals of Statistics 41(3), 1462–1484) 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-sided kernels of unbounded support are more accurate, in this article, 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. Central limit theorems for properly de-biased estimators 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.