A study of the equivalence of inference results in the contexts of true and misspecified multivariate general linear models

Ruixia Yuan; Bo Jiang; Yongge Tian; Ruixia Yuan; Bo Jiang; Yongge Tian

doi:10.3934/math.20231069

AIMS Mathematics

2023, Volume 8, Issue 9: 21001-21021. doi: 10.3934/math.20231069

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Research article

A study of the equivalence of inference results in the contexts of true and misspecified multivariate general linear models

1.
College of Business and Economics, Shanghai Business School, Shanghai 201400, China
2.
College of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005, China

Received: 03 May 2023 Revised: 11 June 2023 Accepted: 15 June 2023 Published: 30 June 2023
MSC : 62F12, 62F30, 62J10

In practical applications of regression models, we may meet with the situation where a true model is misspecified in some other forms due to certain unforeseeable reasons, so that estimation and statistical inference results obtained under the true and misspecified regression models are not necessarily the same, and therefore, it is necessary to compare these results and to establish certain links between them for the purpose of reasonably explaining and utilizing the misspecified regression models. In this paper, we propose and investigate some comparison and equivalence analysis problems on estimations and predictions under true and misspecified multivariate general linear models. We first give the derivations and presentations of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of unknown parametric matrices under a true multivariate general linear model and its misspecified form. We then derive a variety of necessary and sufficient conditions for the BLUPs/BLUEs under the two competing models to be equal using a series of highly-selective formulas and facts associated with ranks, ranges and generalized inverses of matrices, as well as block matrix operations.
- block matrix,
- BLUE,
- BLUP,
- misspecified model,
- MGLM,
- rank formula
Citation: Ruixia Yuan, Bo Jiang, Yongge Tian. A study of the equivalence of inference results in the contexts of true and misspecified multivariate general linear models[J]. AIMS Mathematics, 2023, 8(9): 21001-21021. doi: 10.3934/math.20231069

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Abstract

In practical applications of regression models, we may meet with the situation where a true model is misspecified in some other forms due to certain unforeseeable reasons, so that estimation and statistical inference results obtained under the true and misspecified regression models are not necessarily the same, and therefore, it is necessary to compare these results and to establish certain links between them for the purpose of reasonably explaining and utilizing the misspecified regression models. In this paper, we propose and investigate some comparison and equivalence analysis problems on estimations and predictions under true and misspecified multivariate general linear models. We first give the derivations and presentations of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of unknown parametric matrices under a true multivariate general linear model and its misspecified form. We then derive a variety of necessary and sufficient conditions for the BLUPs/BLUEs under the two competing models to be equal using a series of highly-selective formulas and facts associated with ranks, ranges and generalized inverses of matrices, as well as block matrix operations.

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