Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution - Filippo Palombi; Simona Toti; Romina Filippini (Journal of Probability and Statistics)
We relate the matrix SB of the second moments of a spherically truncated normal multivariate to its full covariance matrix Σ and present an algorithm to invert the relation and reconstruct Σ from SB. While the eigenvectors of Σ are left invariant by the truncation, its eigenvalues are nonuniformly damped. We show that the eigenvalues of Σ can be reconstructed from their truncated counterparts via a fixed point iteration, whose convergence we prove analytically. The procedure requires the computation of multidimensional Gaussian integrals over an Euclidean ball, for which we extend a numerical technique, originally proposed by Ruben in 1962, based on a series expansion in chi-square distributions. In order to study the feasibility of our approach, we examine the convergence rate of some iterative schemes on suitably chosen ensembles of Wishart matrices. We finally discuss the practical difficulties arising in sample space and outline a regularization of the problem based on perturbation theory.