Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution

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 d...

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Bibliographic Details
Published in:Journal of probability and statistics 2017-01, Vol.2017 (2017), p.1-24
Main Authors: Palombi, Filippo, Filippini, Romina, Toti, Simona
Format: Article
Language:English
Subjects:
Convergence
Covariance matrix
Eigenvalues
Integral equations
Integrals
Invariants
Mathematical analysis
Regularization
Series expansion
Statistics
Symmetry
Variables
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Summary: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.
ISSN:1687-952X
1687-9538
DOI:10.1155/2017/6579537