Home > ELA > Vol. 33 (2018)
Article Title
On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices
Keywords
Banded Toeplitz matrix, Projection, Convex cone, Estimation, Covariance structure.
Abstract
We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.
Recommended Citation
Filipiak, Katarzyna; Markiewicz, Augustyn; Mieldzioc, Adam; and Sawikowska, Aneta.
(2018),
"On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices",
Electronic Journal of Linear Algebra,
Volume 33, pp. 74-82.
DOI: https://doi.org/10.13001/1081-3810.3750
Abstract
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