Estimation of Population Linear Spectral Statistics by Marchenko--Pastur Inversion

A new method of estimating population linear spectral statistics from high-dimensional data is introduced. When the dimension $d$ grows with the sample size $n$ such that $\frac{d}{n}\to c>0$, the proposed method is the first with proven convergence rate of $\mathcal{O}(n^{\epsilon -1})$ for any $\epsilon >0$ in a general nonparametric setting. For Gaussian data, a CLT for the estimation error with normalization factor $n$ is shown.