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 $\textit{WeSpeR}$ algorithm, which estimates the spectral density and retrieves the true spectral covariance spectrum. Empirical tests confirm the good properties of the $\textit{WeSpeR}$ algorithm.