Truncated Estimators for a Precision Matrix

• Published in: Mathematical methods of statistics Vol. 33; no. 1; pp. 12 - 25
• Main Authors: Haddouche, Anis M., Fourdrinier, Dominique
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-03-2024; Springer Nature B.V
• Subjects: Approximation; Canonical forms; Covariance matrix; Estimators; Mathematical analysis; Mathematics and Statistics; Regression models; Statistical Theory and Methods; Statistics; high–dimensional statistics; Data-based loss; precision matrix; truncated estimators; statistical decision theory
• ISSN: 1066-5307; 1934-8045
• DOI: 10.3103/S1066530724700029

In this paper, we estimate the precision matrix
of a Gaussian multivariate linear regression model through its canonical form
where
and
are respectively an
and an
matrices. This problem is addressed under the data-based loss function
, where
estimates
, for any ordering of
and
, in a unified approach. We derive estimators which, besides the information contained in the sample covariance matrix
, use the information contained in the sample mean
. We provide conditions for which these estimators improve over the usual estimators
where
is a positive constant and
is the Moore-Penrose inverse of
. Thanks to the role of
, such estimators are also improved by their truncated version.