WeSpeR: Population spectrum retrieval and spectral density estimation of weighted sample covariance

The spectrum of the weighted sample covariance shows a asymptotic non random behavior when the dimension grows with the number of samples. In this setting, we prove that the asymptotic spectral distribution F of the weighted sample covariance has a continuous density on \mathbb{R}^*. We address then the practical problem of numerically finding this density. We propose a procedure to compute it, to determine the support of F and define an efficient grid on it. We use this procedure to design the WeSpeR algorithm, which estimates the spectral density and retrieves the true spectral covariance spectrum. Empirical tests confirm the good properties of the WeSpeR algorithm.