Mathematics – Statistics Theory
Scientific paper
2012-02-08
Mathematics
Statistics Theory
26 pages, 7 figures
Scientific paper
In this paper we relate the matrix $S_B$ of the second moments of a spherically truncated normal multivariate to its full covariance matrix $\Sigma$ and present an algorithm to invert the relation and reconstruct $\Sigma$ from $S_B$. While the eigenvectors of $\Sigma$ are left invariant by the truncation, its eigenvalues are non-uniformly damped. We show that the eigenvalues of $\Sigma$ 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 a 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.
Filippini Romina
Palombi Filippo
Toti Simona
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