Truncated Estimators for a Precision Matrix

Abstract

In this paper, we estimate the precision matrix \({\Sigma}^{-1}\) of a Gaussian multivariate linear regression model through its canonical form \(({Z}^{T},{U}^{T})^{T}\) where \(Z\) and \(U\) are respectively an \(m\times p\) and an \(n\times p\) matrices. This problem is addressed under the data-based loss function \(\textrm{tr}\ [({\hat{\Sigma}}^{-1}-{\Sigma}^{-1})S]^{2}\), where \({\hat{\Sigma}}^{-1}\) estimates \({\Sigma}^{-1}\), for any ordering of \(m,n\) and \(p\), in a unified approach. We derive estimators which, besides the information contained in the sample covariance matrix \(S={U}^{T}U\), use the information contained in the sample mean \(Z\). We provide conditions for which these estimators improve over the usual estimators \(a{S}^{+}\) where \(a\) is a positive constant and \({S}^{+}\) is the Moore-Penrose inverse of \(S\). Thanks to the role of \(Z\), such estimators are also improved by their truncated version.