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Models with commutative orthogonal block structure: a general condition for commutativity.

C Santos1,2, C Nunes3, C Dias4,2

  • 1Department of Mathematics and Physical Sciences, Polytechnic Institute of Beja, Beja, Portugal.

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|June 16, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces commutative orthogonal block structure (COBS) models, a specialized class of linear mixed models. COBS models ensure that least squares estimators are the best linear unbiased estimators for estimable vectors.

Keywords:
U-matricesbest linear unbiased estimatorsmixed modelsmodels with commutative orthogonal block structure

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Area of Science:

  • Statistics
  • Linear Models
  • Mixed Models

Background:

  • Orthogonal block structure (OBS) models are linear mixed models with specific variance-covariance matrices.
  • OBS models offer desirable properties for estimable vectors and variance components.
  • Least squares estimators can be the best linear unbiased estimators (BLUE) in OBS models under certain conditions.

Purpose of the Study:

  • To define and explore a more restricted class of OBS models, termed commutative orthogonal block structure (COBS) models.
  • To establish a commutativity condition for achieving BLUE properties with least squares estimators.
  • To introduce U-matrices as a tool for defining this commutativity condition.

Main Methods:

  • Defining linear mixed models with variance-covariance matrices as linear combinations of orthogonal projection matrices.
  • Introducing the concept of orthogonal block structure (OBS).
  • Deriving the commutativity condition between projection matrices for the mean vector and variance-covariance structure.
  • Utilizing U-matrices to characterize the commutativity condition.

Main Results:

  • Identified commutative orthogonal block structure (COBS) models as a subset of OBS models.
  • Demonstrated that COBS models allow least squares estimators to be BLUE for estimable vectors.
  • Presented a commutativity condition involving U-matrices for defining COBS models.

Conclusions:

  • COBS models provide a framework for enhanced estimation properties in linear mixed models.
  • The commutativity condition, facilitated by U-matrices, is key to achieving optimal estimation.
  • This work contributes to the theoretical understanding and application of structured covariance matrices in statistical modeling.