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314642

Quadratic forms in Normal Variates Under Ridge Regression. By: Abdul-Mordy Hamed Azzam

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Last updated: 05 Jan 2025

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Abstract

This paper concentrates on studying the quadratic forms in normal variates which appear when testing linear statistical hypothesis under ridge regression with positive non-stochastic biased factors k1, k2, …, kp . Except for the correction factor nȳ2, it is shown that all other quadratic forms are not independent and do not follow central or non-central x2 distributions. Hence the classical F statistics are irrelevant under ridge. The results of Hoerl and Kennard (1990) are obtained as special cases when the biased factors are all positive and equal. Moreover, the classical ordinary least squares results are also obtained as special cases when all the biased factors are set to zero.

DOI

10.21608/esju.1997.314642

Keywords

Ordinary Least Squares (OLS), Ordinary Ridge (OR), Generalized Ridge (GR), Sum of Squares of Regression under OLS (SSRols), Sum of Squares of Errors under OLS (SSEols), Sum of Squares of Interaction under OLS (SSIols), Sum of Squares of Regression under Ridge (SSRr), Sum of Squares of Errors under Ridge (SSEr), Sum of Squares of Interaction under Ridge (SSIr), Mean Square Error (MSE), Orthogonal Projection Operator (OPO), Range space of a matrix (Ṟ(.)], Null space of a matrix [Ṉ(.)], trace of a matrix (tr(.)], Canonical Parametrization

Volume

41

Article Issue

2

Related Issue

43146

Issue Date

1997-12-01

Publish Date

1997-12-01

Page Start

95

Page End

109

Print ISSN

0542-1748

Online ISSN

2786-0086

Link

https://esju.journals.ekb.eg/article_314642.html

Detail API

https://esju.journals.ekb.eg/service?article_code=314642

Order

2

Type

Original Article

Type Code

1,914

Publication Type

Journal

Publication Title

The Egyptian Statistical Journal

Publication Link

https://esju.journals.ekb.eg/

MainTitle

Quadratic forms in Normal Variates Under Ridge Regression. By: Abdul-Mordy Hamed Azzam

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Article

Created At

28 Dec 2024