Efficient Computation of Limit Spectra of Sample Covariance Matrices

Consider an $n \times p$ data matrix $X$ whose rows are independently sampled from a population with covariance $\Sigma$. When $n,p$ are both large, the eigenvalues of the sample covariance matrix are substantially different from those of the true covariance. Asymptotically, as $n,p \to \infty$ with $p/n \to \gamma$, there is a deterministic mapping from the population spectral distribution (PSD) to the empirical spectral distribution (ESD) of the eigenvalues. The mapping is characterized by a fixed-point equation for the Stieltjes transform. We propose a new method to compute numerically the output ESD from an arbitrary input PSD. Our method, called Spectrode, finds the support and the density of the ESD to high precision; we prove this for finite discrete distributions. In computational experiments it outperforms existing methods by several orders of magnitude in speed and accuracy. We apply Spectrode to compute expectations and contour integrals of the ESD. These quantities are often central in applications of random matrix theory (RMT). We illustrate that Spectrode is directly useful in statistical problems, such as estimation and hypothesis testing for covariance matrices. Our proposal may make it more convenient to use asymptotic RMT in aspects of high-dimensional data analysis.