Eigen-inference 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 d/n → c>0, the introduced method is the first to provably achieve eigen-inference with fast convergence rates of O(nε-1) for any ε > 0 in the general non-parametric setting. This is achieved though a novel Marchenko-Pastur inversion formula, which may also be formulated as a semi-explicit solution to the Marchenko-Pastur equation.