A Gauss-Markov Theorem for Infinite Dimensional Regression Models with Possibly Singular Covariance

Consider the linear model z = Bx + v where x is a vector, B a linear operator, and v is a random variable with mean 0 and covariance$V^2 = \operatorname{cov} (\mathbf{v, v})$. A linear functional (g, ·) is called a best linear unbiased estimator for c if (i) B * g = c, and (ii) |V2g|2is as small as...

Full description

Saved in:
Bibliographic Details
Published inSIAM journal on applied mathematics Vol. 37; no. 2; pp. 257 - 260
Main Author Morley, T. D.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.10.1979
Subjects
Covariance
Gauss Markov theorem
Hilbert space
Hilbert spaces
Linear transformations
Mathematical theorems
Mathematical vectors
Random variables
Regression analysis
Unbiased estimators
Online AccessGet full text
ISSN0036-1399
1095-712X
DOI10.1137/0137016

Cover

More Information
Summary:Consider the linear model z = Bx + v where x is a vector, B a linear operator, and v is a random variable with mean 0 and covariance$V^2 = \operatorname{cov} (\mathbf{v, v})$. A linear functional (g, ·) is called a best linear unbiased estimator for c if (i) B * g = c, and (ii) |V2g|2is as small as possible, subject to (i). The classical Gauss-Markov theorem gives a formula for the best linear unbiased estimator in finite dimensions, under the condition that V2is invertible. We extend the Gauss-Markov theorem to Hilbert space without the condition that the covariance is invertible.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0036-1399
1095-712X
DOI:10.1137/0137016