Finite sample Bernstein-von Mises theorems for functionals and spectral projectors of the covariance matrix

Abstract

We demonstrate that a prior influence on the posterior distribution of covariance matrix vanishes as sample size grows. The assumptions on a prior are explicit and mild. The results are valid for a finite sample and admit the dimension $p$ growing with the sample size $n$. We exploit the described fact to derive the finite sample Bernstein-von Mises theorem for functionals of covariance matrix (e.g. eigenvalues) and to find the posterior distribution of the Frobenius distance between spectral projector and empirical spectral projector. This can be useful for constructing sharp confidence sets for the true value of the functional or for the true spectral projector.

32 pages, submitted version

References in corpus (5)

• High Dimensional Statistical Inference and Random Matrices
• Normal approximation and concentration of spectral projectors of sample covariance
• Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes
• Principal component analysis for big data
• Bootstrap confidence sets for spectral projectors of sample covariance