Statistical proofs of some matrix inequalities. (English) Zbl 0977.15007
Matrix algebra is extensively used in the study of linear models and multivariate analysis. During recent years, there have been a number of papers where statistical results are used to prove some matrix theorems.
In this paper, a number of matrix results are proved using some properties of Fisher information and covariance matrices. A unified approach is provided through the use of Schur complements. The statistical results used in the proof are briefly reviewed in the introduction section.
The paper contains the sections of Cauchy-Schwarz and related inequalities; A matrix equality; Schur and Kronecker product of matrices; Milne’s inequality; Convexity of some matrix functions; Carlen’s superadditivity of Fisher information; and Inequalities on principal submatrices of an \(nnd\) matrix.
In this paper, a number of matrix results are proved using some properties of Fisher information and covariance matrices. A unified approach is provided through the use of Schur complements. The statistical results used in the proof are briefly reviewed in the introduction section.
The paper contains the sections of Cauchy-Schwarz and related inequalities; A matrix equality; Schur and Kronecker product of matrices; Milne’s inequality; Convexity of some matrix functions; Carlen’s superadditivity of Fisher information; and Inequalities on principal submatrices of an \(nnd\) matrix.
Reviewer: Yueh-er Kuo (Knoxville)
MSC:
| 15A45 | Miscellaneous inequalities involving matrices |
| 62B10 | Statistical aspects of information-theoretic topics |
| 62G05 | Nonparametric estimation |
Keywords:
Carlen’s inequality; generalized inverse; Kronecker product; Milne’s inequality; Schur product; convexity; Fisher information; covariance matrices; Schur complementsReferences:
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