Abstract: This paper considers the problem of estimating the population spectral distribution from a sample covariance matrix when its dimension is large. We generalize the contour-integral based method in Mestre (2008) and present a local moment estimation procedure. Compared with the original, the new procedure can be applied successfully to models where the asymptotic clusters of sample eigenvalues generated by different population eigenvalues are not all separate. The proposed estimates are proved to be consistent. Numerical results illustrate the implementation of the estimation procedure and demonstrate its efficiency in various cases.

Key words and phrases: Empirical spectral distribution, large covariance matrix, moment estimation, population spectral distribution, Stieltjes transform.