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Journal of Probability and Statistics
Volume 2017, Article ID 6579537, 24 pages
https://doi.org/10.1155/2017/6579537
Research Article

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

1Istituto Nazionale di Statistica (ISTAT), Via Cesare Balbo 16, 00184 Rome, Italy
2Italian Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Via Enrico Fermi 45, 00044 Frascati, Italy

Correspondence should be addressed to Filippo Palombi; ti.aene@ibmolap.oppilif

Received 7 July 2016; Revised 7 October 2016; Accepted 12 October 2016; Published 10 January 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Filippo Palombi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We relate the matrix of the second moments of a spherically truncated normal multivariate to its full covariance matrix and present an algorithm to invert the relation and reconstruct from . While the eigenvectors of are left invariant by the truncation, its eigenvalues are nonuniformly damped. We show that the eigenvalues of 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.