| Physical description |
xiii, 293 pages ; 24 cm. |
| Series |
Statistics, textbooks and monographs ; v. 91. |
|
Statistics, textbooks and monographs ; v. 91.
|
| Notes |
Includes indexes. |
| Bibliography |
Bibliography: pages 275-276. |
| Contents |
1 Gauss and Gauss-Jordan Methods: Aspects of Computer Programming 1 -- 1.2 Gauss's Method 3 -- 1.3 Gauss's Method with Row Interchanges 8 -- 1.4 Gauss-Jordan Method 10 -- 1.5 Arithmetical Cost 11 -- 1.6 Efficient Programming 13 -- 1.7 Computer Representation of Numbers 15 -- 1.8 A Measure of Computational Accuracy 16 -- 1.9 Gauss's Method with Integer Coefficients 17 -- 2 Matrix Analysis of Gauss's Method: The Cholesky and Doolittle Decompositions 23 -- 2.1 Matrix Representation 23 -- 2.2 Matrix Multiplication 26 -- 2.3 Matrix Inversion 27 -- 2.4 Elementary Matrices 30 -- 2.5 Matrix Analysis of Gauss's Method 32 -- 2.6 Determinant 34 -- 2.7 Doolittle's L[subscript 0]U[subscript *] Decomposition 35 -- 2.8 U'[subscript 0]DU[subscript 0] Decomposition 37 -- 2.9 Cholesky's U'U Decomposition 38 -- 2.10 Horst's Method 40 -- 3 Linear Algebraic Model: The Method of Averages and the Method of Least Squares 45 -- 3.1 Linear Algebraic Model 45 -- 3.2 Method of Averages 46 -- 3.3 Method of Least Squares 49 -- 3.4 Accuracy 50 -- 3.5 Empirical Condition Number 51 -- 3.6 Longley's Test Problem 52 -- 4 Cauchy-Bienayme, Laplace, and Schmidt Procedures 59 -- 4.1 Cauchy-Bienayme Procedure 59 -- 4.3 Laplace Orthogonalization Procedure 66 -- 4.4 Schmidt Orthogonalization Procedure 68 -- 4.5 Comparison of the Schmidt and Laplace Procedures 69 -- 4.6 Laplace's Procedure with Column Interchanges 71 -- 4.7 Uniqueness of the U'[subscript 0]DU[subscript 0] Decomposition 72 -- 4.8 Partial Orthogonalization and Scaling 73 -- 5 Householder's Procedure 81 -- 5.2 Householder's Transformation Matrix 84 -- 5.3 Comparison of the Householder and Laplace Procedures 89 -- 5.4 Further Remarks on Householder's Procedure 90 -- 5.5 Maindonald's Variant of Householder's Procedure 91 -- 5.6 Householder's Procedure with Column Interchanges 93 -- 6 Givens's Procedure 97 -- 6.1 Givens Transformation Matrix 97 -- 6.3 A Revised Version of Givens's Procedure 101 -- 6.4 Partial Orthogonalization 103 -- 6.5 Square Root Free Variants of Givens's Procedure 106 -- 6.6 Gentleman's Procedure 109 -- 6.7 Weighted Least Squares Estimator 112 -- 6.8 Deleting Observations 113 -- 6.9 Imposing Constraints 115 -- 6.10 Stirling's Procedure 118 -- 6.11 Generalized Givens Transformations with Integer Coefficients 120 -- 7 Updating the qu Decomposition 127 -- 7.1 Adding Rows 127 -- 7.2 Deleting Rows 129 -- 7.3 Adding Columns 131 -- 7.4 Deleting Columns 132 -- 7.5 Permuting Columns 133 -- 7.6 All Possible Regressions 135 -- 7.7 Adding Dummy Rows and Dummy Columns 137 -- 8 Pseudo-Random Numbers 141 -- 8.1 Data Precision 141 -- 8.2 Multiplicative Congruential Pseudo-Random Number Generators 142 -- 8.3 Uniformly Distributed Pseudo-Random Numbers 145 -- 8.4 Mean and Variance 146 -- 8.5 Normally Distributed Pseudo-Random Numbers 147 -- Project: A Simulation Study 149 -- 9 Standard Linear Model 153 -- 9.1 Linear Statistical Model 153 -- 9.2 Expectation and Variance of a Random Variable 155 -- 9.3 Standard Linear Model 156 -- 9.4 Expectation and Variance of an Estimator 158 -- 9.5 Expectation and Variance of [beta] 159 -- 9.6 Least Squares Estimator of [sigma superscript 2] 161 -- 9.7 Expected Results of the Simulation Study 162 -- Project: A Least Squares Computer Program 164 -- 10 Condition Numbers 167 -- 10.1 Theoretical Condition Numbers 167 -- 10.2 Empirical Condition Numbers 170 -- 10.3 Perturbations of the Full Data Set 170 -- 11 Instrumental Variable Estimators 173 -- 11.1 Instrumental Variable Estimator 173 -- 11.2 A Nonsymmetric Variant of Householder's Procedure 174 -- 11.3 Nonsymmetric Householder Transformation Matrices 177 -- 11.4 Expectation and Variance of an Instrumental Variable Estimator 179 -- 11.6 Computing Instrumental Variable Estimators by Householder's Procedure 182 -- 12 Generalized Least Squares Estimation 185 -- 12.1 Grouped Data 185 -- 12.2 Three Linear Models 188 -- 12.3 Weighted Least Squares and Generalized Least Squares Estimators 189 -- 12.4 Elementary Computational Procedures 191 -- 12.5 Generalized Least Squares Estimation by Householder Transformations 192 -- 12.6 A Reexamination of Householder's Procedure 195 -- 12.7 Reverse Cholesky Decomposition 196 -- 12.8 Generalized Least Squares Estimation by Givens Transformations 198 -- 12.10 Updating the Generalized Least Squares Estimates 202 -- 12.11 Special Methods for the Reverse Cholesky Decomposition 206 -- 12.12 Generalizations of the Generalized Least Squares Problem 208 -- 12.13 Estimation in Rank Deficient Models 211 -- 13 Iterative Solutions of Linear and Nonlinear Least Squares Problems 217 -- 13.1 Iterative Refinement of the Generalized Least Squares Estimator 217 -- 13.2 Iterative Refinement of the Ordinary Least Squares Estimator 219 -- 13.3 Iterative Solution of Nonlinear Least Squares Problems 223 -- 14 Canonical Expressions for the Least Squares Estimators and Test Statistics 229 -- 14.1 Canonical Form of the Standard Linear Model 229 -- 14.2 Unbiased Estimators of [beta] and [sigma superscript 2] 231 -- 14.3 Minimum Variance Unibased Linear Estimator of [beta] 232 -- 14.4 A Statistical Test for Deleted Regressors 234 -- 14.5 Distributional Assumptions 235 -- 14.7 A Statistical Test for Additional Observations 239 -- Project: A Simulation Study of the Deletion Test 241 -- 15 Traditional Expressions for the Least Squares Updating Formulas and Test Statistics 245 -- 15.1 Adding Regressors 245 -- 15.3 Inverse of a Partitioned Matrix 251 -- 15.4 Adding Observations 252 -- 15.6 Deleting Observations 256 -- 15.7 Imposing Constraints 257 -- 15.8 A Statistical Test for Linear Constraints 259 -- 16 Least Squares Estimation Subject to Linear Equality Constraints 265 -- 16.1 Inconsistent Constraints 265 -- 16.2 General Form the Constrained Least Squares Estimator 267 -- 16.3 A Particular Form of the Constrained Least Squares Estimator 268 -- 16.4 Constrained Minimum Variance Unbiased Linear Estimator of [beta] 269 -- 16.5 Traditional Expressions for the Constrained Least Squares Estimator 271 -- 16.6 Distribution of the Test for Linear Constraints 272. |
| Summary |
Presenting numerous algorithms in a simple algebraic form so that the reader can easily translate them into any computer language, this volume gives details of several methods for obtaining accurate least squares estimates. It explains how these estimates may be updated as new information becomes available and how to test linear hypotheses. Linear Least Squares Computations features many structured exercises that guide the reader through the available algorithms, plus a glossary of commonly used terms and a bibliography of supplementary reading ... collects "ancient" and modern results on linear least squares computations in a convenient single source ... develops the necessary matrix algebra in the context of multivariate statistics ... only makes peripheral use of concepts such as eigenvalues and partial differentiation ... interprets canonical forms employed in computation ... discusses many variants of the Gauss, Laplace-Schmidt, Givens, and Householder algorithms ... and uses an empirical approach for the appraisal of algorithms. Linear Least Squares Computations serves as an outstanding reference for industrial and applied mathematicians, statisticians, and econometricians, as well as a text for advanced undergraduate and graduate statistics, mathematics, and econometrics courses in computer programming, linear regression analysis, and applied statistics. |
| Subject |
Least squares.
|
| ISBN |
0824776615 |
|
