A matrix analysis of BLMBPs under a general linear model and its transformation

Li Gong; Bo Jiang; Li Gong; Bo Jiang

doi:10.3934/math.2024090

AIMS Mathematics

2024, Volume 9, Issue 1: 1840-1860. doi: 10.3934/math.2024090

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A matrix analysis of BLMBPs under a general linear model and its transformation

Li Gong ¹,
Bo Jiang ^{2
,
,}

1.
Department of Primary School Education, Fuxin Higher Vocational College, Fuxin, Liaoning, China
2.
College of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong, China

Received: 03 November 2023 Revised: 28 November 2023 Accepted: 04 December 2023 Published: 15 December 2023
MSC : 15A09, 62H12, 62J05

This paper is concerned with the relationships between best linear minimum biased predictors (BLMBPs) in the context of a general linear model (GLM) and its transformed general linear models (TGLMs). We shall establish a mathematical procedure by means of some exact and analytical tools in matrix theory that were developed in recent years. The coverage includes constructing a general vector composed of all unknown parameters in the context of a GLM and its TGLMs, deriving the exact expressions of the BLMBPs through the technical use of analytical solutions of a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and discussing a variety of theoretical performances and properties of the BLMBPs. We also give a series of characterizations of relationships between BLMBPs under a given GLM and its TGLMs.
- general linear model,
- transformed general linear model,
- LMBP,
- BLMBP,
- relationships,
- rank
Citation: Li Gong, Bo Jiang. A matrix analysis of BLMBPs under a general linear model and its transformation[J]. AIMS Mathematics, 2024, 9(1): 1840-1860. doi: 10.3934/math.2024090

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Abstract

This paper is concerned with the relationships between best linear minimum biased predictors (BLMBPs) in the context of a general linear model (GLM) and its transformed general linear models (TGLMs). We shall establish a mathematical procedure by means of some exact and analytical tools in matrix theory that were developed in recent years. The coverage includes constructing a general vector composed of all unknown parameters in the context of a GLM and its TGLMs, deriving the exact expressions of the BLMBPs through the technical use of analytical solutions of a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and discussing a variety of theoretical performances and properties of the BLMBPs. We also give a series of characterizations of relationships between BLMBPs under a given GLM and its TGLMs.

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