{2}-Inverses and Their Statistical Application

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 {12}-Inverses: The Null Augmented Mappings. - 2. 4 {15}-Inverses: The Nonnull Augmented Mappings. - 2. 5 Construction of Moore-Penrose Type Generalized Inverses. - 2. 6 A Geometric Representation of {2}-Inverses. - 2. 7 {15}-Inverses and Projections. - 2. 8 {15}-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 {12}-Inverses and Sets of Nonestimable Constraints. - 4. 4 Bott-Duffin Inverses and Constrained LSS's. - 4. 5 {15}-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. Language: English