Estimation of the population spectral distribution from a large dimensional sample covariance matrix

This paper introduces a new method to estimate the spectral distribution of a population covariance matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marčenko–Pastur equation, originally defined in the complex plane, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation matrix of S&P 500 daily stock returns is also given.