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

• The paper introduces a new method for the estimation of the population spectral distribution from high-dimensional data.
• The estimation method is based on a generalization of the Marcenko-Pastur equation on the real line.
• Simulation experiments show that the new estimator outperforms some existing estimators in the literature.
• A data set of S&P 500 Stock returns is analyzed..

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.