Optimal Unbiased Estimation of Variance Components

Inhaltsverzeichnis

• One: The Basic Model and the Estimation Problem.
• 1.1 Introduction.
• 1.2 An Example.
• 1.3 The Matrix Formulation.
• 1.4 The Estimation Criteria.
• 1.5 Properties of the Criteria.
• 1.6 Selection of Estimation Criteria.
• Two: Basic Linear Technique.
• 2.1 Introduction.
• 2.2 The vec and mat Operators.
• 2.3 Useful Properties of the Operators.
• Three: Linearization of the Basic Model.
• 3.1 Introduction.
• 3.2 The First Linearization.
• 3.3 Calculation of var(y).
• 3.4 The Second Linearization of the Basic Model.
• 3.5 Additional Details of the Linearizations.
• Four: The Ordinary Least Squares Estimates.
• 4.1 Introduction.
• 4.2 The Ordinary Least Squares Estimates: Calculation.
• 4.3 The Inner Structure of the Linearization.
• 4.4 Estimable Functions of the Components.
• 4.5 Further OLS Facts.
• Five: The Seely-Zyskind Results.
• 5.1 Introduction.
• 5.2 The General Gauss-Markov Theorem: Some History and Motivation.
• 5.3 The General Gauss-Markov Theorem: Preliminaries.
• 5.4 The General Gauss-Markov Theorem: Statement and Proof.
• 5.5 The Zyskind Version of the Gauss-Markov Theorem.
• 5.6 The Seely Condition for Optimal unbiased Estimation.
• Six: The General Solution to Optimal Unbiased Estimation.
• 6.1 Introduction.
• 6.2 A Full Statement of the Problem.
• 6.3 The Lehmann-Scheffé Result.
• 6.4 The Two Types of Closure.
• 6.5 The General Solution.
• 6.6 An Example.
• Seven: Background from Algebra.
• 7.1 Introduction.
• 7.2 Groups, Rings, Fields.
• 7.3 Subrings and Ideals.
• 7.4 Products in Jordan Rings.
• 7.5 Idempotent and Nilpotent Elements.
• 7.6 The Radical of an Associative or Jordan Algebra.
• 7.7 Quadratic Ideals in Jordan Algebras.
• Eight: The Structure of Semisimple Associative and Jordan Algebras.
• 8.1 Introduction.
• 8.2 The First Structure Theorem.
• 8.3 Simple Jordan Algebras.
• 8.4 SimpleAssociative Algebras.
• Nine: The Algebraic Structure of Variance Components.
• 9.1 Introduction.
• 9.2 The Structure of the Space of Optimal Kernels.
• 9.3 The Two Algebras Generated by Sp(?2).
• 9.4 Quadratic Ideals in Sp(?2).
• 9.5 Further Properties of the Space of Optimal Kernels.
• 9.6 The Case of Sp(?2) Commutative.
• 9.7 Examples of Mixed Model Structure Calculations: The Partially Balanced Incomplete Block Designs.
• Ten: Statistical Consequences of the Algebraic Structure Theory.
• 10.1 Introduction.
• 10.2 The Jordan Decomposition of an Optimal Unbiased Estimate.
• 10.3 Non-Negative Unbiased Estimation.
• Concluding Remarks.
• References.