Methods of multivariate statistics:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
2002
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| Schriftenreihe: | Wiley series in probability and statistics
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| Online-Zugang: | Publisher description Table of contents Inhaltsverzeichnis |
| Beschreibung: | XIX, 697 S. graph. Darst. |
| ISBN: | 0471223816 |
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| 100 | 1 | |a Srivastava, Muni S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Methods of multivariate statistics |c M. S. Srivastava |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 2002 | |
| 300 | |a XIX, 697 S. |b graph. Darst. | ||
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| 490 | 0 | |a Wiley series in probability and statistics | |
| 650 | 4 | |a Multivariate analysis | |
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| 689 | 0 | 0 | |a Multivariate Analyse |0 (DE-588)4040708-1 |D s |
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Datensatz im Suchindex
| _version_ | 1804138169842532352 |
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| adam_text | Contents
Abbreviations and Notations xv
Preface xvii
1 Multivariate Methods: An Overview 1
1.1 Introduction 1
1.2 One- and Two-Sample Problems 3
1.3 Detecting Change-Point 6
1.4 Data from More Than Two Populations: (MANOVA) 8
1.5 Classification, Discrimination, and Closeness 11
1.6 Multivariate Regression Analysis 12
1.7 Growth Curve Models 17
1.8 Principal Component Analysis 18
1.9 Appendix 19
2 Multivariate Normal Distributions 20
2.1 Introduction 20
2.2 Some Notation 20
2.3 Estimation of Mean and Covariance 24
2.4 Definition of Multivariate Normal 28
2.5 Properties of Multivariate Normal
Distributions 29
2.6 Maximum Likelihood Estimates of ft and £ 37
2.7 Some Results on Quadratic Forms 38
2.8 Appendix 41
2.8.1 Moment-generating Function 41
2.8.2 Jacobian of Transformations 42
2.8.3 Square Root of a Positive Definite Matrix 43
2.8.4 Matrix Differentiation 44
2.8.5 SAS Computational Procedures 45
2.9 Problems 50
3 Outliers Detection and Normality Check 57
3.1 Introduction 57
3.2 Detecting an Outlier in Multivariate Data 58
3.3 Assessing Normality of the Data 60
3.4 Assessing Univariate Normality 60
vii
viii CONTENTS
3.4.1 Kolmogorov s Test 61
3.4.2 Shapiro-Wilk Test 63
3.4.3 A Graphical Method 65
3.4.4 Transformations to Achieve Normality 65
3.5 Assessing Multivariate Normality 68
3.5.1 Small s Graphical Method 69
3.5.2 Srivastava s Graphical Method 70
3.5.3 A Test for Multivariate Normality 71
3.5.4 Test Based on Skewness and Kurtosis Statistic 73
3.5.5 Transformations to Achieve Normality 74
3.6 Examples 74
4 Inference on Location-Hotelling s T2 89
4.1 Introduction 89
4.2 Univariate Testing Problems 89
4.2.1 One-Sample Student s t-Test 89
4.2.2 Two-Sample Student s *-Test 91
4.3 Multivariate One-Sample Testing 92
4.3.1 Multivariate One-Sample Problem 92
4.3.2 Confidence Regions 96
4.3.3 Fixed-Width Confidence Regions 101
4.3.4 Some Remarks 102
4.3.5 Roy s Union-Intersection Method 103
4.3.6 One-sided Tests 104
4.4 Multivariate Two-Sample Testing 109
4.4.1 Two-Sample Problem: Equal Covariance 109
4.4.2 One-Sided Tests with Equal Covariance 112
4.4.3 Paired T2-Test 114
4.4.4 One-Sided paired T2-Test 116
4.4.5 Behrens-Fisher Problem: Unequal Covariance 118
4.5 A Test for a Subvector 122
4.5.1 One-Sample Case 123
4.5.2 Two-Sample Case 125
4.6 Tests for Detecting a Change in Mean 129
4.6.1 An Estimate of the Change Point 130
4.6.2 Two More Tests for a Shift in the Mean 131
4.7 Tests for Linear Contrasts 133
4.7.1 Testing the Equality of Mean Components 134
4.7.2 One-Sided Tests for Equality of Means 138
4.8 Appendix 140
4.8.1 Analysis of Variance Model 140
4.8.2 Derivation of LRT for /x = fj,0 142
4.9 Problems 144
CONTENTS ix
5 Repeated Measures 152
5.1 Introduction 152
5.2 Intraclass Correlation Model 152
5.3 Repeated Measures: One-Sample Case 154
5.3.1 Testing Equality of Means 154
5.3.2 ANOVA Table in Terms of Sufficient Statistic 155
5.3.3 Confidence Intervals 156
5.3.4 Examples 157
5.3.5 Justification for the Results in (5.3.2) 161
5.4 Inter and Intraclass Correlation Model 162
5.4.1 Model for Familial Data with Children Only 162
5.4.2 Model for the Familial Data with Mother 163
5.5 Hereditary Coefficient 165
5.5.1 Solution of Testing Problem in (i) 166
5.5.2 Solution of Testing Problem in (ii) 167
5.5.3 Solution for Testing Problem in (iii) 167
5.5.4 Equal Sample Size Case 168
5.6 Split-Plot and MANOVA Designs 169
5.7 Problems 171
6 Multivariate Analysis of Variance 178
6.1 Introduction 178
6.2 Completely Randomized Design 178
6.2.1 Derivation of the Likelihood Ratio Test 186
6.2.2 Some Other Possible Tests 186
6.3 Randomized Complete Block Design 187
6.4 Latin Square Design 194
6.5 Factorial Experiments 198
6.6 Analysis of Covariance 203
6.7 Appendix 208
6.8 Problems 211
7 Profile Analysis 219
7.1 Introduction 219
7.2 Profile Analysis of Two Groups 220
7.2.1 Tests for Similarity of Profiles 222
7.2.2 Tests of the Level Hypothesis 223
7.2.3 Test for the Condition Variation 225
7.3 General Case of J Groups 232
7.4 Problems 242
8 Classification and Discrimination 246
8.1 Introduction 246
8.2 Classifying into Two Known Normals 247
8.3 Classifying into Two Normals,
Means Unknown 249
8.3.1 Estimates of ei and e2 250
8.3.2 An Example 251
x CONTENTS
8.4 Classifying into Two Unknown Normals 252
8.4.1 Estimates of the Errors of Misclassification 253
8.4.2 An Example 254
8.5 Classifying into k Normals 255
8.6 The Canonical Variates Method 257
8.6.1 An Example 259
8.6.2 Proof of (8.6.3) 261
8.7 A Test for Reduction in Number of Variables 262
8.8 Stepwise Discriminant Analysis 263
8.9 Reduced Rank Discrimination 264
8.10 Classification with Covariates 264
8.11 Classifying into Two Normals with Unequal Covariance 265
8.12 Problems 265
9 Multivariate Regression 269
9.1 Introduction 269
9.2 Multiple Linear Regression 269
9.3 Multivariate Linear Regression Model 277
9.3.1 Prediction, Residual, and Covariance Estimate 281
9.3.2 A Measure of Fit 281
9.4 Tests and Confidence Intervals 282
9.5 Comparing Regression Models 302
9.6 Testing Bilinear Hypotheses 305
9.7 Selection of Independent Variables 310
9.8 Prediction by Principal Component Method 312
9.8.1 Centering the Regression Model 313
9.8.2 Centering and Scaling the Regression Model 314
9.8.3 Scaled Version of the Regression Model 316
9.8.4 Principal Component Method 316
9.8.5 PC Method in Multivariate Regression Model 319
9.8.6 How to Choose Between the LSE and PC Method 322
9.9 Ridge Regression Estimators 327
9.9.1 Bayes Estimator of Regression Parameters 331
9.9.2 Empirical Bayes Estimator of c or A 333
9.9.3 Ridge-Principal Component Estimators 334
9.9.4 Empirical Bayes Multivariate Ridge Estimator 336
9.10 Shrinkage or Stein Type of Estimators 337
9.11 Random Design Matrix 340
9.11.1 Least Squares Estimator for Random Design Matrix 340
9.11.2 Testing £ , = 0 341
9.11.3 Testing CE = 0 342
9.12 Detecting Outliers and Assessing Normality 344
9.13 Appendix 347
9.13.1 Distributional Results for Matrix of Normals 347
9.13.2 Maximization of R2a in Subsection 9.3.1 348
9.13.3 Variance Stabilizing Transformations 348
9.13.4 Proof of Z y(0) = Z y in Section 9.8 349
9.13.5 Bias and Mean Square Error in PC Method 350
CONTENTS xi
9.13.6 Effect on Estimates of ^ 352
9.13.7 Bias in Prediction 353
9.13.8 Mallow s C,-Statistic 355
9.13.9 Akaike s Criterion 358
9.14 Problems 359
10 Growth Curve Models 365
10.1 Introduction 365
10.2 One-Sample GCM-Polynomial Regression 365
10.2.1 Test of the Adequacy of the Model 366
10.2.2 Estimates and Confidence Intervals for rp 369
10.2.3 Test of the General Linear Hypotheses 369
10.3 Generalized MANOVA-GCM Model 373
10.3.1 Test for the Adequacy of the Model 375
10.3.2 MLE of the Parameters V and E 376
10.3.3 LRT for Testing a Submatrix of V to Be Zero 376
10.3.4 LRT for Testing tp = 0 377
10.3.5 LRT for General Hypotheses 378
10.3.6 Simultaneous Confidence Intervals 378
10.4 Testing for an Outlier in
Growth Curve Models 389
10.5 Problems 392
11 Principal Component Analysis 397
11.1 Introduction 397
11.2 PC Analysis Based on the Covariance Matrix 397
11.3 PC Analysis Based on the Sample Covariance 402
11.3.1 LRT for the Equality of the Last p - k
Eigenvalues of E 408
11.3.2 Some Asymptotic Distributions 408
11.3.3 Effect of Units of Measurement 409
11.4 Uses of Principal Components 410
11.5 PC Based on Sample Correlation Matrix 413
11.6 Problems 421
12 Factor Analysis 428
12.1 Introduction 428
12.2 Model 430
12.3 Communality, Variance of a Factor, and Total Variance 435
12.4 Comparison with Principal Components 436
12.5 Estimation of Parameters 438
12.5.1 Maximum Likelihood Estimates 438
12.5.2 Principal Factor Analysis 439
12.6 Choosing the Number of Factors 441
12.7 Problem of Negative Estimates of the Variances (*,) 443
12.8 Selection of Loadings and Factors 445
12.8.1 Factor Rotation 445
12.8.2 Varimax Rotation 446
xii CONTENTS
12.8.3 Quartimax Rotation 452
12.8.4 Oblimin Rotations 453
12.9 Factor Scores 454
12.10Examples 454
12.11Appendix 468
12.11.1 Verification of (12.1.1) 468
12.11.2Derivative of the Maximum Likelihood Solution 468
12.11.3 An Iterative Solution without Normality
Assumption 470
12.12Problems 471
13 Inference on Covariance Matrices 479
13.1 Introduction 479
13.2 A Test for E = Eo 479
13.2.1 Statement of the Problem 480
13.2.2 LRT and Its Asymptotic Distribution 480
13.2.3 A Test for E = / 481
13.2.4 Derivation of the LRT 481
13.3 A Test for Sphericity 481
13.3.1 Statement of the Problem 481
13.3.2 LRT and Its Asymptotic Distribution 482
13.3.3 Derivation of LRT 482
13.3.4 Some Comments 482
13.4 A Test for an Intraclass Correlation Model 483
13.4.1 Statement of the Problem 484
13.4.2 LRT and Its Asymptotic Distribution 484
13.4.3 Derivation of LRT 485
13.5 A Test for Equicorrelation 486
13.5.1 Statement of the Problem 487
13.5.2 A Test 487
13.6 A Test for Zero Correlation 489
13.6.1 Statement of the Problem 489
13.6.2 LRT and Its Asymptotic Distribution 489
13.7 A Test for Equality of Covariances 489
13.7.1 Statement of the Problem 489
13.7.2 LRT and Its Asymptotic Distribution 490
13.8 A Test for Independence 492
13.9 Testing for S = I and i = 0 494
13.9.1 Statement of the Problem 494
13.9.2 LRT and Its Asymptotic Distribution 494
13-10A Test for S = r2l and n = 0 494
13.10.1 Statement of the Problem 494
13.10.2 LRT and Its Asymptotic Distribution 495
13.11Equality of Mean Vectors and Covariances 495
13.12Problems 497
CONTENTS xiii
14 Correlations 501
14.1 Introduction 501
14.2 Correlation Between Two Random Variables 502
14.2.1 Estimating p When au = 022 = a2 504
14.2.2 Estimating p When au = 722 = 1 and p 0 (or 0) .... 504
14.3 Estimating p in the Intraclass Correlation Model 505
14.4 Matrix of Sample Correlations 506
14.5 Partial Correlations 507
14.6 Multiple Correlation 509
14.7 Canonical Correlations 512
14.7.1 Sample Canonical Correlation 515
14.7.2 Some Tests 516
14.7.3 Likelihood Ratio Test for Independence 517
14.7.4 An Example 518
14.8 Problems 520
15 Missing Observations: General Case 528
15.1 Introduction 528
15.2 MLE: Bivariate Case 528
15.2.1 Iterative Solution of the Likelihood Equations 530
15.2.2 Testing of Hypothesis: /x = /i0 532
15.3 General p 534
15.4 EM Algorithm 538
15.5 Appendix 541
16 Missing Observations: Monotone Sample 551
16.1 Introduction 551
16.2 Tests and Estimates: Bivariate Case 552
16.3 General p: Estimation 556
16.4 General p: Testing for Location 560
16.5 Testing Equality of the Mean Components 565
16.6 Testing Equality of Two Mean Vectors 566
16.7 Problems 571
17 Bootstrapping 572
17.1 Introduction 572
17.2 Bootstrap Methods 573
17.3 Bootstrapping from Residuals 575
17.4 Confidence Interval for a
Parametric Function 577
17.5 Bootstrapping in Multiple Regression Model 580
17.6 Testing About the Variance 584
17.7 Testing that the Mean Vector /x = Mo 586
17.8 Testing the Equality of Two Mean Vectors 590
17.9 Bootstrapping Multivariate Regression 594
17.10Testing that the Covariance Is a
Specified Matrix 596
Xiv CONTENTS
17.11Testing the Equality of Two Covariance
Matrices 599
17.12Appendix 604
17.12.1 Asymptotic Distribution Theory 604
17.12.2 Consistency of Sample cdf 605
18 Imputing Missing Data 607
18.1 Introduction 607
18.2 Regression Model with Missing Observations 611
18.2.1 Bootstrap Estimate of the Covariance of bi* 613
18.2.2 Bootstrap Distribution 615
18.3 Multivariate Data with Missing Observations 615
18.3.1 Description of Missing Observations 617
18.3.2 Imputing Missing Values and Estimates
of Parameters 618
18.3.3 Bootstrap Estimate of the Covariance of jx* 619
18.3.4 An Example 620
18.4 Bootstrap Method in Multivariate
Incomplete Data 621
18.5 Appendix: Covariance of ft* 623
A Some Results on Matrices 626
A.I Notation and Definitions 626
A.2 Matrix Operations 628
A.3 Determinants 630
A.3.1 Cofactors of a Square Matrix 631
A.3.2 Minor, Principal Minor, and Trace of a Matrix 631
A.3.3 Eigenvalues and Eigenvectors 632
A.4 Rank of a Matrix 633
A.5 Inverse of a Nonsingular Matrix 634
A.6 Generalized Inverse of a Matrix 635
A.7 Idempotent Matrices 636
A.8 Positive Definite and Positive Semidefinite Matrices 636
A.9 Some Inequalities 637
A.10 Problems 638
B Tables 640
Bibliography 664
|
| adam_txt |
Contents
Abbreviations and Notations xv
Preface xvii
1 Multivariate Methods: An Overview 1
1.1 Introduction 1
1.2 One- and Two-Sample Problems 3
1.3 Detecting Change-Point 6
1.4 Data from More Than Two Populations: (MANOVA) 8
1.5 Classification, Discrimination, and Closeness 11
1.6 Multivariate Regression Analysis 12
1.7 Growth Curve Models 17
1.8 Principal Component Analysis 18
1.9 Appendix 19
2 Multivariate Normal Distributions 20
2.1 Introduction 20
2.2 Some Notation 20
2.3 Estimation of Mean and Covariance 24
2.4 Definition of Multivariate Normal 28
2.5 Properties of Multivariate Normal
Distributions 29
2.6 Maximum Likelihood Estimates of ft and £ 37
2.7 Some Results on Quadratic Forms 38
2.8 Appendix 41
2.8.1 Moment-generating Function 41
2.8.2 Jacobian of Transformations 42
2.8.3 Square Root of a Positive Definite Matrix 43
2.8.4 Matrix Differentiation 44
2.8.5 SAS Computational Procedures 45
2.9 Problems 50
3 Outliers Detection and Normality Check 57
3.1 Introduction 57
3.2 Detecting an Outlier in Multivariate Data 58
3.3 Assessing Normality of the Data 60
3.4 Assessing Univariate Normality 60
vii
viii CONTENTS
3.4.1 Kolmogorov's Test 61
3.4.2 Shapiro-Wilk Test 63
3.4.3 A Graphical Method 65
3.4.4 Transformations to Achieve Normality 65
3.5 Assessing Multivariate Normality 68
3.5.1 Small's Graphical Method 69
3.5.2 Srivastava's Graphical Method 70
3.5.3 A Test for Multivariate Normality 71
3.5.4 Test Based on Skewness and Kurtosis Statistic 73
3.5.5 Transformations to Achieve Normality 74
3.6 Examples 74
4 Inference on Location-Hotelling's T2 89
4.1 Introduction 89
4.2 Univariate Testing Problems 89
4.2.1 One-Sample Student's t-Test 89
4.2.2 Two-Sample Student's *-Test 91
4.3 Multivariate One-Sample Testing 92
4.3.1 Multivariate One-Sample Problem 92
4.3.2 Confidence Regions 96
4.3.3 Fixed-Width Confidence Regions 101
4.3.4 Some Remarks 102
4.3.5 Roy's Union-Intersection Method 103
4.3.6 One-sided Tests 104
4.4 Multivariate Two-Sample Testing 109
4.4.1 Two-Sample Problem: Equal Covariance 109
4.4.2 One-Sided Tests with Equal Covariance 112
4.4.3 Paired T2-Test 114
4.4.4 One-Sided paired T2-Test 116
4.4.5 Behrens-Fisher Problem: Unequal Covariance 118
4.5 A Test for a Subvector 122
4.5.1 One-Sample Case 123
4.5.2 Two-Sample Case 125
4.6 Tests for Detecting a Change in Mean 129
4.6.1 An Estimate of the Change Point 130
4.6.2 Two More Tests for a Shift in the Mean 131
4.7 Tests for Linear Contrasts 133
4.7.1 Testing the Equality of Mean Components 134
4.7.2 One-Sided Tests for Equality of Means 138
4.8 Appendix 140
4.8.1 Analysis of Variance Model 140
4.8.2 Derivation of LRT for /x = fj,0 142
4.9 Problems 144
CONTENTS ix
5 Repeated Measures 152
5.1 Introduction 152
5.2 Intraclass Correlation Model 152
5.3 Repeated Measures: One-Sample Case 154
5.3.1 Testing Equality of Means 154
5.3.2 ANOVA Table in Terms of Sufficient Statistic 155
5.3.3 Confidence Intervals 156
5.3.4 Examples 157
5.3.5 Justification for the Results in (5.3.2) 161
5.4 "Inter" and Intraclass Correlation Model 162
5.4.1 Model for Familial Data with Children Only 162
5.4.2 Model for the Familial Data with Mother 163
5.5 Hereditary Coefficient 165
5.5.1 Solution of Testing Problem in (i) 166
5.5.2 Solution of Testing Problem in (ii) 167
5.5.3 Solution for Testing Problem in (iii) 167
5.5.4 Equal Sample Size Case 168
5.6 Split-Plot and MANOVA Designs 169
5.7 Problems 171
6 Multivariate Analysis of Variance 178
6.1 Introduction 178
6.2 Completely Randomized Design 178
6.2.1 Derivation of the Likelihood Ratio Test 186
6.2.2 Some Other Possible Tests 186
6.3 Randomized Complete Block Design 187
6.4 Latin Square Design 194
6.5 Factorial Experiments 198
6.6 Analysis of Covariance 203
6.7 Appendix 208
6.8 Problems 211
7 Profile Analysis 219
7.1 Introduction 219
7.2 Profile Analysis of Two Groups 220
7.2.1 Tests for Similarity of Profiles 222
7.2.2 Tests of the Level Hypothesis 223
7.2.3 Test for the Condition Variation 225
7.3 General Case of J Groups 232
7.4 Problems 242
8 Classification and Discrimination 246
8.1 Introduction 246
8.2 Classifying into Two Known Normals 247
8.3 Classifying into Two Normals,
Means Unknown 249
8.3.1 Estimates of ei and e2 250
8.3.2 An Example 251
x CONTENTS
8.4 Classifying into Two Unknown Normals 252
8.4.1 Estimates of the Errors of Misclassification 253
8.4.2 An Example 254
8.5 Classifying into k Normals 255
8.6 The Canonical Variates Method 257
8.6.1 An Example 259
8.6.2 Proof of (8.6.3) 261
8.7 A Test for Reduction in Number of Variables 262
8.8 Stepwise Discriminant Analysis 263
8.9 Reduced Rank Discrimination 264
8.10 Classification with Covariates 264
8.11 Classifying into Two Normals with Unequal Covariance 265
8.12 Problems 265
9 Multivariate Regression 269
9.1 Introduction 269
9.2 Multiple Linear Regression 269
9.3 Multivariate Linear Regression Model 277
9.3.1 Prediction, Residual, and Covariance Estimate 281
9.3.2 A Measure of Fit 281
9.4 Tests and Confidence Intervals 282
9.5 Comparing Regression Models 302
9.6 Testing Bilinear Hypotheses 305
9.7 Selection of Independent Variables 310
9.8 Prediction by Principal Component Method 312
9.8.1 Centering the Regression Model 313
9.8.2 Centering and Scaling the Regression Model 314
9.8.3 Scaled Version of the Regression Model 316
9.8.4 Principal Component Method 316
9.8.5 PC Method in Multivariate Regression Model 319
9.8.6 How to Choose Between the LSE and PC Method 322
9.9 Ridge Regression Estimators 327
9.9.1 Bayes Estimator of Regression Parameters 331
9.9.2 Empirical Bayes Estimator of c or A 333
9.9.3 Ridge-Principal Component Estimators 334
9.9.4 Empirical Bayes Multivariate Ridge Estimator 336
9.10 Shrinkage or Stein Type of Estimators 337
9.11 Random Design Matrix 340
9.11.1 Least Squares Estimator for Random Design Matrix 340
9.11.2 Testing £ , = 0 341
9.11.3 Testing CE = 0 342
9.12 Detecting Outliers and Assessing Normality 344
9.13 Appendix 347
9.13.1 Distributional Results for Matrix of Normals 347
9.13.2 Maximization of R2a in Subsection 9.3.1 348
9.13.3 Variance Stabilizing Transformations 348
9.13.4 Proof of Z'y(0) = Z'y in Section 9.8 349
9.13.5 Bias and Mean Square Error in PC Method 350
CONTENTS xi
9.13.6 Effect on Estimates of ^ 352
9.13.7 Bias in Prediction 353
9.13.8 Mallow's C,-Statistic 355
9.13.9 Akaike's Criterion 358
9.14 Problems 359
10 Growth Curve Models 365
10.1 Introduction 365
10.2 One-Sample GCM-Polynomial Regression 365
10.2.1 Test of the Adequacy of the Model 366
10.2.2 Estimates and Confidence Intervals for rp 369
10.2.3 Test of the General Linear Hypotheses 369
10.3 Generalized MANOVA-GCM Model 373
10.3.1 Test for the Adequacy of the Model 375
10.3.2 MLE of the Parameters V and E 376
10.3.3 LRT for Testing a Submatrix of V to Be Zero 376
10.3.4 LRT for Testing tp = 0 377
10.3.5 LRT for General Hypotheses 378
10.3.6 Simultaneous Confidence Intervals 378
10.4 Testing for an Outlier in
Growth Curve Models 389
10.5 Problems 392
11 Principal Component Analysis 397
11.1 Introduction 397
11.2 PC Analysis Based on the Covariance Matrix 397
11.3 PC Analysis Based on the Sample Covariance 402
11.3.1 LRT for the Equality of the Last p - k
Eigenvalues of E 408
11.3.2 Some Asymptotic Distributions 408
11.3.3 Effect of Units of Measurement 409
11.4 Uses of Principal Components 410
11.5 PC Based on Sample Correlation Matrix 413
11.6 Problems 421
12 Factor Analysis 428
12.1 Introduction 428
12.2 Model 430
12.3 Communality, Variance of a Factor, and Total Variance 435
12.4 Comparison with Principal Components 436
12.5 Estimation of Parameters 438
12.5.1 Maximum Likelihood Estimates 438
12.5.2 Principal Factor Analysis 439
12.6 Choosing the Number of Factors 441
12.7 Problem of Negative Estimates of the Variances (*,) 443
12.8 Selection of Loadings and Factors 445
12.8.1 Factor Rotation 445
12.8.2 Varimax Rotation 446
xii CONTENTS
12.8.3 Quartimax Rotation 452
12.8.4 Oblimin Rotations 453
12.9 Factor Scores 454
12.10Examples 454
12.11Appendix 468
12.11.1 Verification of (12.1.1) 468
12.11.2Derivative of the Maximum Likelihood Solution 468
12.11.3 An Iterative Solution without Normality
Assumption 470
12.12Problems 471
13 Inference on Covariance Matrices 479
13.1 Introduction 479
13.2 A Test for E = Eo 479
13.2.1 Statement of the Problem 480
13.2.2 LRT and Its Asymptotic Distribution 480
13.2.3 A Test for E = / 481
13.2.4 Derivation of the LRT 481
13.3 A Test for Sphericity 481
13.3.1 Statement of the Problem 481
13.3.2 LRT and Its Asymptotic Distribution 482
13.3.3 Derivation of LRT 482
13.3.4 Some Comments 482
13.4 A Test for an Intraclass Correlation Model 483
13.4.1 Statement of the Problem 484
13.4.2 LRT and Its Asymptotic Distribution 484
13.4.3 Derivation of LRT 485
13.5 A Test for Equicorrelation 486
13.5.1 Statement of the Problem 487
13.5.2 A Test 487
13.6 A Test for Zero Correlation 489
13.6.1 Statement of the Problem 489
13.6.2 LRT and Its Asymptotic Distribution 489
13.7 A Test for Equality of Covariances 489
13.7.1 Statement of the Problem 489
13.7.2 LRT and Its Asymptotic Distribution 490
13.8 A Test for Independence 492
13.9 Testing for S = I and \i = 0 494
13.9.1 Statement of the Problem 494
13.9.2 LRT and Its Asymptotic Distribution 494
13-10A Test for S = r2l and n = 0 494
13.10.1 Statement of the Problem 494
13.10.2 LRT and Its Asymptotic Distribution 495
13.11Equality of Mean Vectors and Covariances 495
13.12Problems 497
CONTENTS xiii
14 Correlations 501
14.1 Introduction 501
14.2 Correlation Between Two Random Variables 502
14.2.1 Estimating p When au = 022 = a2 504
14.2.2 Estimating p When au = 722 = 1 and p 0 (or 0) . 504
14.3 Estimating p in the Intraclass Correlation Model 505
14.4 Matrix of Sample Correlations 506
14.5 Partial Correlations 507
14.6 Multiple Correlation 509
14.7 Canonical Correlations 512
14.7.1 Sample Canonical Correlation 515
14.7.2 Some Tests 516
14.7.3 Likelihood Ratio Test for Independence 517
14.7.4 An Example 518
14.8 Problems 520
15 Missing Observations: General Case 528
15.1 Introduction 528
15.2 MLE: Bivariate Case 528
15.2.1 Iterative Solution of the Likelihood Equations 530
15.2.2 Testing of Hypothesis: /x = /i0 532
15.3 General p 534
15.4 EM Algorithm 538
15.5 Appendix 541
16 Missing Observations: Monotone Sample 551
16.1 Introduction 551
16.2 Tests and Estimates: Bivariate Case 552
16.3 General p: Estimation 556
16.4 General p: Testing for Location 560
16.5 Testing Equality of the Mean Components 565
16.6 Testing Equality of Two Mean Vectors 566
16.7 Problems 571
17 Bootstrapping 572
17.1 Introduction 572
17.2 Bootstrap Methods 573
17.3 Bootstrapping from Residuals 575
17.4 Confidence Interval for a
Parametric Function 577
17.5 Bootstrapping in Multiple Regression Model 580
17.6 Testing About the Variance 584
17.7 Testing that the Mean Vector /x = Mo 586
17.8 Testing the Equality of Two Mean Vectors 590
17.9 Bootstrapping Multivariate Regression 594
17.10Testing that the Covariance Is a
Specified Matrix 596
Xiv CONTENTS
17.11Testing the Equality of Two Covariance
Matrices 599
17.12Appendix 604
17.12.1 Asymptotic Distribution Theory 604
17.12.2 Consistency of Sample cdf 605
18 Imputing Missing Data 607
18.1 Introduction 607
18.2 Regression Model with Missing Observations 611
18.2.1 Bootstrap Estimate of the Covariance of bi* 613
18.2.2 Bootstrap Distribution 615
18.3 Multivariate Data with Missing Observations 615
18.3.1 Description of Missing Observations 617
18.3.2 Imputing Missing Values and Estimates
of Parameters 618
18.3.3 Bootstrap Estimate of the Covariance of jx* 619
18.3.4 An Example 620
18.4 Bootstrap Method in Multivariate
Incomplete Data 621
18.5 Appendix: Covariance of ft* 623
A Some Results on Matrices 626
A.I Notation and Definitions 626
A.2 Matrix Operations 628
A.3 Determinants 630
A.3.1 Cofactors of a Square Matrix 631
A.3.2 Minor, Principal Minor, and Trace of a Matrix 631
A.3.3 Eigenvalues and Eigenvectors 632
A.4 Rank of a Matrix 633
A.5 Inverse of a Nonsingular Matrix 634
A.6 Generalized Inverse of a Matrix 635
A.7 Idempotent Matrices 636
A.8 Positive Definite and Positive Semidefinite Matrices 636
A.9 Some Inequalities 637
A.10 Problems 638
B Tables 640
Bibliography 664 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T22:34:03Z |
| indexdate | 2024-07-09T21:23:55Z |
| institution | BVB |
| isbn | 0471223816 |
| language | English |
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| spelling | Srivastava, Muni S. Verfasser aut Methods of multivariate statistics M. S. Srivastava New York [u.a.] Wiley 2002 XIX, 697 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd rswk-swf Multivariate Analyse (DE-588)4040708-1 s DE-604 http://www.loc.gov/catdir/description/wiley036/2002728174.html Publisher description http://www.loc.gov/catdir/toc/wiley031/2002728174.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016846557&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Srivastava, Muni S. Methods of multivariate statistics Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd |
| subject_GND | (DE-588)4040708-1 |
| title | Methods of multivariate statistics |
| title_auth | Methods of multivariate statistics |
| title_exact_search | Methods of multivariate statistics |
| title_exact_search_txtP | Methods of multivariate statistics |
| title_full | Methods of multivariate statistics M. S. Srivastava |
| title_fullStr | Methods of multivariate statistics M. S. Srivastava |
| title_full_unstemmed | Methods of multivariate statistics M. S. Srivastava |
| title_short | Methods of multivariate statistics |
| title_sort | methods of multivariate statistics |
| topic | Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd |
| topic_facet | Multivariate analysis Multivariate Analyse |
| url | http://www.loc.gov/catdir/description/wiley036/2002728174.html http://www.loc.gov/catdir/toc/wiley031/2002728174.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016846557&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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