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Tables of Contents for An Introduction to Multivariate Statistical Analysis
Chapter/Section Title
Page #
Page Count
CHAPTER 1 Introduction
1
5
1.1. Multivariate Statistical Analysis
1
2
1.2. The Multivariate Normal Distribution
3
3
CHAPTER 2 The Multivariate Normal Distribution
6
53
2.1. Introduction
6
1
2.2. Notions of Multivariate Distributions
7
7
2.3. The Multivariate Normal Distribution
14
10
2.4. The Distribution of Linear Combinations of Normally Distributed Variates; Independence of Variates; Marginal Distributions
24
11
2.5. Conditional Distributions and Multiple Correlation Coefficient
35
8
2.6. The Characteristic Function; Moments
43
7
Problems
50
9
CHAPTER 3 Estimation of the Mean Vector and the Covariance Matrix
59
43
3.1. Introduction
59
1
3.2. The Maximum Likelihood Estimators of the Mean Vector and the Covariance Matrix
60
8
3.3. The Distribution of the Sample Mean Vector; Inference Concerning the Mean When the Covariance Matrix Is Known
68
9
3.4. Theoretical Properties of Estimators of the Mean Vector
77
9
3.5. Improved Estimation of the Mean
86
10
Problems
96
6
CHAPTER 4 The Distributions and Uses of Sample Correlation Coefficients
102
54
4.1. Introduction
102
1
4.2. Correlation Coefficient of a Bivariate Sample
103
22
4.3. Partial Correlation Coefficients; Conditional Distributions
125
9
4.4. The Multiple Correlation Coefficient
134
15
Problems
149
7
CHAPTER 5 The Generalized T^2-Statistic
156
39
5.1. Introduction
156
1
5.2. Derivation of the Generalized T^2-Statistic and Its Distribution
157
7
5.3. Uses of the T^2-Statistic
164
9
5.4. The Distribution of T^2 Under Alternative Hypotheses; The Power Function
173
2
5.5. The Two-Sample Problem with Unequal Covariance Matrices
175
6
5.6. Some Optimal Properties of the T^2-Test
181
9
Problems
190
5
CHAPTER 6 Classification of Observations
195
49
6.1. The Problem of Classification
195
1
6.2. Standards of Good Classification
196
3
6.3. Procedures of Classification into One of Two Populations with Known Probability Distributions
199
5
6.4. Classification into One of Two Known Multivariate Normal Populations
204
4
6.5. Classification into One of Two Multivariate Normal Populations When the Parameters Are Estimated
208
9
6.6. Probabilities of Misclassification
217
7
6.7. Classification into One of Several Populations
224
4
6.8. Classification into One of Several Multivariate Normal Populations
228
3
6.9. An Example of Classification into One of Several Multivariate Normal Populations
231
3
6.10. Classification into One of Two Known Multivariate Normal Populations with Unequal Covariance Matrices
234
7
Problems
241
3
CHAPTER 7 The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance
244
41
7.1. Introduction
244
1
7.2. The Wishart Distribution
245
7
7.3. Some Properties of the Wishart Distribution
252
5
7.4. Cochran's Theorem
257
2
7.5. The Generalized Variance
259
7
7.6. Distribution of the Set of Correlation Coefficients When the Population Covariance Matrix Is Diagonal
266
2
7.7. The Inverted Wishart Distribution and Bayes Estimation of the Covariance Matrix
268
5
7.8. Improved Estimation of the Covariance Matrix
273
6
Problems
279
6
CHAPTER 8 Testing the General Linear Hypothesis; Multivariate Analysis of Variance
285
91
8.1. Introduction
285
2
8.2. Estimators of Parameters in Multivariate Linear Regression
287
5
8.3. Likelihood Ratio Criteria for Testing Linear Hypotheses About Regression Coefficients
292
6
8.4. The Distribution of the Likelihood Ratio Criterion When the Hypothesis Is True
298
13
8.5. An Asymptotic Expansion of the Distribution of the Likelihood Ratio Criterion
311
10
8.6. Other Criteria for Testing the Linear Hypothesis
321
12
8.7. Tests of Hypotheses About Matrices of Regression Coefficients and Confidence Regions
333
5
8.8. Testing Equality of Means of Several Normal Distributions with Common Covariance Matrix
338
4
8.9. Multivariate Analysis of Variance
342
7
8.10. Some Optimal Properties of Tests
349
20
Problems
369
7
CHAPTER 9 Testing Independence of Sets of Variates
376
28
9.1. Introduction
376
1
9.2. The Likelihood Ratio Criterion for Testing Independence of Sets of Variates
376
5
9.3. The Distribution of the Likelihood Ratio Criterion When the Null Hypothesis Is True
381
4
9.4. An Asymptotic Expansion of the Distribution of the Likelihood Ratio Criterion
385
2
9.5. Other Criteria
387
2
9.6. Step-down Procedures
389
3
9.7. An Example
392
2
9.8. The Case of Two Sets of Variates
394
3
9.9. Admissibility of the Likelihood Ratio Test
397
2
9.10. Monotonicity of Power Functions of Tests of Independence of Sets
399
3
Problems
402
2
CHAPTER 10 Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices
404
47
10.1. Introduction
404
1
10.2. Criteria for Testing Equality of Several Covariance Matrices
405
3
10.3. Criteria for Testing That Several Normal Distributions Are Identical
408
2
10.4. Distributions of the Criteria
410
9
10.5. Asymptotic Expansions of the Distributions of the Criteria
419
3
10.6. The Case of Two Populations
422
5
10.7. Testing the Hypothesis That a Covariance Matrix Is Proportional to a Given Matrix; The Sphericity Test
427
7
10.8. Testing the Hypothesis That a Covariance Matrix Is Equal to a Given Matrix
434
6
10.9. Testing the Hypothesis That a Mean Vector and a Covariance Matrix Are Equal to a Given Vector and Matrix
440
3
443
3
Problems
446
5
CHAPTER 11 Principal Components
451
29
11.1. Introduction
451
1
11.2. Definition of Principal Components in the Population
452
8
11.3. Maximum Likelihood Estimators of the Principal Components and Their Variances
460
2
11.4. Computation of the Maximum Likelihood Estimates of the Principal Components
462
3
11.5. An Example
465
3
11.6. Statistical Inference
468
5
11.7. Testing Hypotheses about the Characteristic Roots of a Covariance Matrix
473
4
Problems
477
3
CHAPTER 12 Canonical Correlations and Canonical Variables
480
41
12.1. Introduction
480
1
12.2. Canonical Correlations and Variates in the Population
481
11
12.3. Estimation of Canonical Correlations and Variates
492
5
12.4. Statistical Inference
497
3
12.5. An Example
500
2
12.6. Linearly Related Expected Values
502
7
12.7. Simultaneous Equations Models
509
10
Problems
519
2
CHAPTER 13 The Distributions of Characteristic Roots and Vectors
521
29
13.1. Introduction
521
1
13.2. The Case of Two Wishart Matrices
522
10
13.3. The Case of One Nonsingular Wishart Matrix
532
6
13.4. Canonical Correlations
538
2
13.5. Asymptotic Distributions in the Case of One Wishart Matrix
540
4
13.6. Asymptotic Distributions in the Case of Two Wishart Matrices
544
4
Problems
548
2
CHAPTER 14 Factor Analysis
550
29
14.1. Introduction
550
1
14.2. The Model
551
6
14.3. Maximum Likelihood Estimators for Random Orthogonal Factors
557
12
14.4. Estimation for Fixed Factors
569
1
14.5. Factor Interpretation and Transformation
570
4
14.6. Estimation for Identification by Specified Zeroes
574
1
14.7. Estimation of Factor Scores
575
1
Problems
576
3
APPENDIX A Matrix Theory
579
30
A.1. Definition of a Matrix and Operations on Matrices
579
8
A.2. Characteristic Roots and Vectors
587
4
A.3. Partitioned Vectors and Matrices
591
5
A.4. Some Miscellaneous Results
596
9
A.5. Gram-Schmidt Orthogonalization and the Solution of Linear Equations
605
4
APPENDIX B Tables
609
34
1. Wilks' Likelihood Criterion: Factors C(p, m, M) to Adjust to X^2(pm) where M = n - p + 1
609
7
2. Tables of Significance Points for the Lawley-Hotelling Trace Test
616
14
3. Tables of Significance Points for the Bartlett-Nanda-Pillai Trace Test
630
4
4. Tables of Significance Points for the Roy Maximum Root Test
634
4
5. Tables of Significance Points for the Modified Likelihood Ratio Test of Equality of Covariance Matrices Based on Equal Sample Sizes
638
1
6. Correction Factors for Significance Points for the Sphericity Test
639
2
7. Significance Points for the Modified Likelihood Ratio Test (Sigma) = (Sigma) (0)
641
2
References
643
24
Index
667