{2}-Inverses and Their Statistical Application (Lecture Notes in Statistics, 47)

Getson, Albert J.; Hsuan, Francis C.

ISBN 10: 0387968490 / ISBN 13: 9780387968490

Editore: Springer, 1988

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Dati bibliografici

• Titolo: {2}-Inverses and Their Statistical Application (Lecture Notes in Statistics, 47)
• Editore: Springer
• Data di pubblicazione: 1988
• Lingua: Inglese
• ISBN 10: 0387968490
• ISBN 13: 9780387968490
• Rilegatura: Brossura
• Condizione: New

Descrizione articolo

Riassunto

A 2-inverse for a matrix A is any matrix G satisfying GAG=G. This book adresses the special, but heretofore ignored, role of 2-inverses in statistics. The relationship among various kinds of generalized inverses and their construction in terms of 2-inverses are explored. The applications discussed include the role of 2-inverses in: canonically representing second-degree polynomials x'Ax+b'x+c, obtaining vectors ß which satisfy K'ß=0 and minimize y'y-Xß(squared), and relating the hypotheses tested by various analytical approaches in generalized linear models. For each topic, the simplification of both the concepts and the notation which results from a 2-inverse approach is stressed and illustrated with examples.

Contenuti

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 of Squares.- 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.