  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
Mar\v{c}enko-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.
