Finite sample Bernsteinvon 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 Bernsteinvon 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.03522
 arXiv:
 arXiv:1712.03522
 Bibcode:
 2017arXiv171203522S
 Keywords:

 Mathematics  Statistics Theory;
 Primary 62F15;
 secondary 62H25
 EPrint:
 32 pages, submitted version