{2}-Inverses and Their Statistical Application - Albert J. Getson

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Sommaire:
I Introduction.- II Time for {2}-Inverses.- 2.0 Introduction.- 2.1 The Three Phase Inversion Procedure.- 2.2 Constrained Inverses.- 2.3 {2}- and {1,2}-Inverses: The Null Augmented Mappings.- 2.4 {1,5}-Inverses: The Nonnull Augmented Mappings.- 2.5 Construction of Moore-Penrose Type Generalized Inverses.- 2.6 A Geometric Representation of {2}-Inverses.- 2.7 {1,5}-Inverses and Projections.- 2.8 {1,5}-Inverses and Solutions to Linear Equations.- 2.9 Decomposition of {2}-Inverses.- 2.10 Spectral Decomposition in Terms of {2}Inverses.- 2.11 Computation of {2}-Inverses.- III {2}-Inverses, Quadratic Forms and Second Degree Polynomials.- 3.0 Introduction.- 3.1 x2 Distribution and Independence of Quadratic Forms and Second Degree Polynomials.- 3.2 Generalized Inverses and Quadratic Forms.- 3.3 {2}-Inverses and x2 Distributed Quadratic Forms.- 3.4 On The Uniqueness of the {2}-Inverse Representation of x2 Distributed Quadratic Forms.- 3.5 A Minimal Sufficient Set of Coefficient Matrices for All X2 Distributed Quadratic Forms.- 3.6 Independence of X2 Distributed Quadratic Forms.- 3.7 A Canonical Representation of Second Degree Polynomials.- 3.8 X2 Distributed Second Degree Polynomials.- 3.9 {2}-Inverses and the Distribution and Independence of Second Degree Polynomials.- IV {2}-Inverses and Least Squares Solutions.- 4.0 Introduction.- 4.1 The Least Squares Problem.- 4.1 Strategies For Obtaining Least Squares Solutions.- 4.3 Symmetric {1,2}-Inverses and Sets of Nonestimable Constraints.- 4.4 Bott-Duffin Inverses and Constrained LSS's.- 4.5 {1,5}-Inverses and LSS's.- 4.6 Relationships Among LSS's.- 4.7 Minimum Norm LSS's.- 4.8 A General Theorem on Constrained LSS's.- 4.9 Residual Sum of Squares and Their Difference.- 4.10 Computing Constrained LSS's and Residual Sum ofSquares.- V {2}-Inverses in Linear Models.- 5.0 Introduction.- 5.1 The Models.- 5.2 The Distribution and Relationships Among the LSS's For the Prameters in Various Models.- 5.3 Hypothesis Testing in Linear Models.- 5.4 Equivalent Numerator Sum of Squares for a Test of Hypothesis.- 5.5 Hypotheses Invariant to Cell Sizes.- 5.6 The R Approach and SAS Type I and Type II Sums of Squares.- 5.7 The R* Approach and the SAS Type III Sum of Squares.- References.