基于噪声观测的高维协方差阵谱推断--及其在微结构噪声下的积分协波动矩阵推断中的应用
On the inference about the spectra of high-dimensional covariance matrix based on noisy observations-with applications to integrated covolatility matrix inference in the presence of microstructure noise
摘要：
在实际应用中，观测值常常被噪声污染，使得得到的样本协方差阵成为信息加噪声型的协方差阵。为了在这种情况下对真实协方差阵的谱进行推断，我们建立了描述真实样本协方差阵的极限谱分布如何依赖于信息加噪声型样本协方差阵的极限谱分布的渐近关系.作为一个应用，我们考虑了基于含有微结构噪声的高频数据的高维扩散过程的积分共波动（ICV）矩阵谱的推断。预平均估计量是一个信息加噪声型的协方差阵，上述结果，连同真实样本协方差阵的谱分布与总体协方差阵的谱分布之间的（广义）联系，使我们能够提出一个两步估计过程，以估计一类扩散过程的ICV的谱分布.进一步提出了另一种估计方法，该方法具有两个理想的性质：消除了微结构噪声的影响，其极限谱分布仅依赖于标准Marčenko-Pastur方程的ICV谱分布。数值研究表明，我们提出的方法可以用来估计的基础协方差阵的频谱噪声观测的基础上。
英文摘要：
In practice, observations are often contaminated by noise, making the resulting sample covariance matrix to be an information-plus-noise-type covariance matrix. Aiming to make inferences about the spectra of the underlying true covariance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (true) sample covariance matrices depends on that of information-plus-noise-type sample covariance matrices. As an application, we consider the inference about the spectra of integrated covolatility (ICV) matrices of high-dimensional diffusion processes based on high-frequency data with microstructure noise. The (slightly modified) pre-averaging estimator is an information-plus-noise-type covariance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of true sample covariance matrices and that of the population covariance matrix, enables us to propose a two-step procedure to estimate the spectral distribution of ICV for a class of diffusion processes. An alternative estimator is further proposed, which possesses two desirable properties: it eliminates the impact of microstructure noise, and its limiting spectral distribution depends only on that of the ICV through the standard Marčenko-Pastur equation. Numerical studies demonstrate that our proposed methods can be used to estimate the spectra of the underlying covariance matrix based on noisy observations.
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