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 $\frac{d}{n} \rightarrow c>0$, the introduced method is the first to provably achieve eigen-inference with fast convergence rates of $\mathcal{O}(n^{\varepsilon-1})$ for any $\varepsilon > 0$ 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.