ON THE ESTIMATION OF INTEGRATED COVARIANCE MATRICES OF HIGH DIMENSIONAL DIFFUSION PROCESSES
ON THE ESTIMATION OF INTEGRATED COVARIANCE MATRICES OF HIGH DIMENSIONAL DIFFUSION PROCESSES
复制标题
DOI：
10.1214/11-aos939
复制
发表时间：
2011-12-01
影响因子：
4.5
通讯作者：
Li, Yingying
中科院分区：
文献类型：
--
作者：
Zheng, Xinghua;Li, Yingying
我们考虑基于高频观测的高维扩散过程的积分协方差（ICV）矩阵估计。我们首先研究最常用的估计量，即已实现协方差（RCV）矩阵。研究表明，在高维情况下，当维数p和观测频率n以相同的速率增长时，RCV的极限谱分布（LSD）不仅取决于目标ICV的共挥发过程，还取决于共挥发过程随时间的变化。建立了加权样本协方差矩阵的Marcenko-Pastur型定理，在此基础上得到了C类扩散过程的RCV的Marcenko-Pastur型定理。结果清楚地说明了共挥发过程的时间变异性如何影响RCV的LSD。我们进一步提出了一种替代估计量，时变调整已实现协方差（TVARCV）矩阵。我们表明，对于C类过程，TVARCV具有通过Marcenko-Pastur方程其LSD仅依赖于目标ICV的LSD的理想性质，因此，特别地，TVARCV可以使用现有算法来恢复ICV的经验谱分布。
We consider the estimation of integrated covariance (ICV) matrices of high dimensional diffusion processes based on high frequency observations. We start by studying the most commonly used estimator, the realized covariance (RCV) matrix. We show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate, the limiting spectral distribution (LSD) of RCV depends on the covolatility process not only through the targeting ICV, but also on how the covolatility process varies in time. We establish a Marcenko-Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marcenko-Pastur type theorem for RCV for a class C of diffusion processes. The results explicitly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class C, the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Marcenko-Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.