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The Theory of linear models and multivariate analysis

Author: Arnold, Steven F. Publisher: Wiley, 1981.Language: EnglishDescription: 475 p. ; 24 cm.ISBN: 0471050652Type of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print QA278 .A76 1981
(Browse shelf)
001159411
Available 001159411
Total holds: 0

Includes bibliographical references and index

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The Theory of Linear Models and Multivariate Analysis Contents 1 Basic Statistical Definitions and Theorems 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. Sufficiency and Completeness, 2 Estimation and Confidence Intervals, 3 Testing Hypotheses, 7 Noncentral Distributions, 9 Invariance in Hypothesis Testing, 11 Invariance in Estimation, 20 Transforming a Model, 25 1 2 Subspaces and Projections 2.1. 2.2. 2.3. Introduction, 32 Subspaces and Bases, 32 Orthogonality and Projections, 34 32 3 Properties of the Multivariate and Spherical Normal Distributions 40 3.1. 3.2. 3.3. 3.4. 3.5. Multivariate Distributions, 40 Nonnegative Definite Mat ri ces and Their Square Roots, 43 The Mahalanobis Distance, 43 The Multivariate Normal Distribution, 45 The Spherical Normal Distribution, 49 55 4 Introduction to Linear Models 4.1. 4.2. Examples, 56 Discussion of the Assumptions for the Linear Model, 58 5 A Sufficient Statistic 5.1. 5.2. 5.3. 5.4. 5.5. The Statistic, 62 Sufficiency, 63 Completeness, 63 The Coordinatized Model, 65 Regression with an Intercept Term, 65 62 xi Contents 6 Estimation 6.1. 6.2. 6.3. 6.4. 6.5. Minimum Variance Unbiased Estimators and Maximum Likelihood Estimators, 68 Best Invariant Estimators, 69 Confidence Intervals and Prediction Intervals, 71 Further Discussion, 73 The Gauss-Markov Theorem, 74 68 7 Testsabout the Mean 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. The F-test, 79 Multiple Regression, 81 Balanced Analysis of Variance, 84 Unbalanced Analysis of Variance, 91 Analysis of Covariance, 103 Optimality of the F-test, 104 Orthogonal Designs, 109 Estimable Functions and Testable Hypotheses, 112 Interaction in the Two-Way Model with No Replication, 116 One-Sided Tests, 118 The Case in which 2 is Known, 119 79 8 Simultaneous Confidence Intervals--Scheffé Type 8.1. 8.2. The Basic Result, 128 Examples, 131 128 9 Tests about the Variance 10 Asymptotic Validity of Procedures under Nonnormal Distributions 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. Definitions and Theorems from Probability Theory, 142 Defining the Model, 143 Discussion of Huber's Condition with Examples, 144 Derivations, 147 Further Discussion, 151 Variance Stabilizing Transformations, 152 Proof of Theorem 10.3, 155 138 141 11 James-Stein and Ridge Estimators 11.1. 11.2. The James-Stein Estimator for µ, 160 The Modified James-Stein Estimator, 165 159 Contents xiii 11.3. 11.4. 11.5. 11.6. 11.7. The Ridge Estimator, 166 An Empirical Bayes Perspective, 167 Sensitivity to Units of Measurement, 169 Other Comments, 170 Estimating ß, 173 12 Inference Based on the Studentized Range Distribution and Bonferroni's Inequality 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. The Studentized Range Distribution, 180 The Studentized Range Test, 181 Simultaneous Confidence Intervals--Tukey Type, 182 Multiple Comparisons, 186 Asymptotic Validity of Studentized Range Procedures, 194 Bonferroni's Inequality, 195 Further Comments, 196 180 13 The Generalized Linear Model 13.1. 13.2. 13.3. The Basic Results, 200 Autocorrelation, 204 Other Results for the Generalized Linear Model, 205 200 14 The Repeated Measures Model 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14.8. 14.9. Statement of the Results, 210 Examples, 214 Some More Linear Algebra, 218 The Basic Result, 220 Sufficiency and Estimation, 223 Hypothesis Testing, 226 Simultaneous Confidence Intervals for Contrasts, 229 Other Results, 230 The Exchangeable Linear Model, 232 209 15 Random Effects and Mixed Models 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. The One-Way Random Effects Model, 245 The Balanced Two-Way Random Effects Model, 253 Balanced Two-Way Mixed Models, 258 Deriving the Random Effects and Mixed Models, 263 The Relationship between the Repeated Measures Model and Certain Mixed Models, 268 Other Results, 269 Proof of Theorem 15.10, 271 242 xiv Contents 16 The Correlation Model 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7. 16.8. 16.9. Sufficiency and Estimation, 278 Testing that y = 0, 280 Testing that yi = 0, 284 Testing Other Hypotheses, 290 Simultaneous Confidence Intervals, 292 Asymptotic Validity of the Procedures, 292 The Best Invariant Estimator of y, 293 Other Optimality Results, 295 Multiple and Partial Correlation Coefficients, 296 276 17 The Distribution Theory for Multivariate Analysis 17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7. Random Matrices, 308 The Matrix Normal Distribution, 310 The Wishart Distribution, 314 An Important Lemma, 317 The Distribution of Hotelling's T2, 319 The Wishart Density Function, 320 Further Comments, 322 308 18 The Multivariate One- and Two-Sample Models--Inference about the Mean Vector 18.1 18.2. 18.3. 18.4. 18.5. 18.6. 18.7. 18.8. 18.9. A Complete Sufficient Statistic and its Distri bution, 327 Minimum Variance Unbiased Estimators, Maximum Likelihood Estimators, and Best Invariant Estimators, 329 James-Stein Estimators for µ, 332 Testing Hypotheses about µ, 335 Simultaneous Confidence Intervals, 339 Asymptotic Validity of Procedures, 340 Generalized Repeated Measures Models, 342 The Two-Sample Model, 343 Other Comments, 344 326 19 The Multivariate Linear Model 19.1. 19.2. 19.3. 19.4. 19.5. Sufficiency and Estimation, 349 Testing the Multivariate Linear Hypothesis, 352 Simultaneous Confidence Intervals for Contrasts, 368 Testing the Generalized Multivariate Linear Hypothesis, 370 Multivariate Regression, 371 348 Contents 19.6. 19.7. 19.8. 19.9. 19.10. 19.11. Multivariate Analysis of Variance, 372 The Generalized Repeated Measures Model, 374 The Asymptotic Validity of the Procedures, 378 James-Stein Estimation, 382 The Growth Curves Model, 385 Another Generalization of the Linear Model, 388 398 20 Discriminant Analysis 20.1. 20.2. 20.3. 20.4. 20.5. 20.6. 20.7. 20.8. Symmetric Hypotheses, 398 The Case of Known Parameters, 400 The Case of Unknown Parameters, 401 Probabilities of Misclassification, 403 A Bayesian Perspective, 405 Testing a Hypothesis about the Discriminant Coefficients, 406 Discrimination among Several Populations, 410 Further Comments, 413 21 Testing Hypotheses about the Covariance Matrix 21.1. 21.2. 21.3. 21.4. 21.5. Testing for Independence, 417 Testing that = 0, 421 Testing the Equality of Covariance Matrices, 425 Testing the Validity of the Ordinary Linear Model, 428 Testing the Validity of the Repeated Measures Model, 430 416 22 Simplifying the Structure of the Covariance Matrix 22.1. 22.2. Principal Components, 435 Canonical Analysis, 440 445 464 471 435 Appendix: Some Matrix Algebra Bibliography Index

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