On the Inference About the Spectral Distribution of High-Dimensional Covariance Matrix Based on High-Frequency Noisy Observations

Abstract

In practice, observations are often contaminated by noise, making the resulting sample co-variance matrix to be a signal-plus-noise sample co-variance matrix. Aiming to make inferences about the spectral distribution of the population co-variance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (signal) sample co-variance matrices depends on that of signal-plus-noise-type sample co-variance matrices.As an application, we consider the inference about the spectral distribution of integrated co-volatility (ICV) matrices of high-dimensional diffusion processes based on high-frequency data with micro-structure noise. The (slightly modified) pre-averaging estimator is a signal-plus-noise sample co-variance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of signal sample co-variance matrices and that of the population co-variance matrix, enables us to propose a two-step procedure to estimate the spectral distribution of ICV for a class of diffusion processes. An alternative approach is further proposed, which possesses several desirable properties: it is more robust, it eliminates the impact of micro-structure noise, and its limiting spectral distribution depends only on that of the ICV through the standard Marcenko-Pastur equation. The performance of the two approaches proposed are examined via simulation studies, under both the synchronous and asynchronous observation settings.