Optimal Unbiased Estimation of Variance Components: Lecture Notes in Statistics, cartea 39

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.