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Author Malley, James D. author Optimal Unbiased Estimation of Variance Components [electronic resource] / by James D. Malley New York, NY : Springer New York, 1986 http://dx.doi.org/10.1007/978-1-4615-7554-2 X, 146 p. 1 illus. online resource

CONTENT

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-Scheffรฉ 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 Simple Associative 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

Mathematics Applied mathematics Engineering mathematics Mathematics Applications of Mathematics

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