 | Fujikoshi Y. Multivariate statistics: high-dimensional and large-sample approximations / Y.Fujikoshi, V.V.Ulyanov, R.Shimizu. - Hoboken: Wiley, 2010. - xviii, 533 p.: ill. - (Wiley series in probability and statistics). - Bibliogr: p.513-526. - Ind.: p.527-533. - ISBN 978-0-470-41169-8
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Preface ................................................... xiii
Glossary of Notation and Abbreviations .................... xvii
1 Multivariate Normal and Related Distributions ................ 1
1.1 Random Vectors .......................................... 1
1.1.1 Mean Vector and Covariance Matrix ................ 1
1.1.2 Characteristic Function and Distribution ......... 5
1.2 Multivariate Normal Distribution ........................ 6
1.2.1 Bivariate Normal Distribution .................... 6
1.2.2 Definition ....................................... 8
1.2.3 Some Properties ................................. 10
1.3 Spherical and Elliptical Distributions ................. 15
1.4 Multivariate Cumulants ................................. 19
Problems .................................................... 24
2 Wishart Distribution ........................................ 29
2.1 Definition ............................................. 29
2.2 Some Basic Properties .................................. 32
2.3 Functions of Wishart Matrices .......................... 36
2.4 Cochran's Theorem ...................................... 39
2.5 Asymptotic Distributions ............................... 40
Problems .................................................... 43
3 Hotelling's T2 and Lambda Statistics ........................ 47
3.1 Hotelling's T2 and Lambda Statistics ................... 47
3.1.1 Distribution of the T2 Statistic ................ 47
3.1.2 Decomposition of T2 and D2 ...................... 49
3.2 Lambda Statistic ....................................... 53
3.2.1 Motivation of the Lambda Statistic .............. 53
3.2.2 Distribution of the Lambda Statistic ............ 55
3.3 Test for Additional Information ........................ 58
3.3.1 Decomposition of the Lambda Statistic ........... 61
Problems .................................................... 64
4 Correlation Coefficients .................................... 69
4.1 Ordinary Correlation Coefficients ...................... 69
4.1.1 Population Correlation .......................... 69
4.1.2 Sample Correlation .............................. 71
4.2 Multiple Correlation Coefficient ....................... 75
4.2.1 Population Multiple Correlation ................. 75
4.2.2 Sample Multiple Correlation ..................... 77
4.3 Partial Correlation .................................... 80
4.3.1 Population Partial Correlation .................. 80
4.3.2 Sample Partial Correlation ...................... 82
4.3.3 Covariance Selection Model ...................... 83
Problems .................................................... 87
5 Asymptotic Expansions for Multivariate Basic Statistics ..... 91
5.1 Edgeworth Expansion and its Validity ................... 91
5.2 Sample Mean Vector and Covariance Matrix ............... 98
5.3 T2 Statistic .......................................... 104
5.3.1 Outlines of Two Methods ........................ 104
5.3.2 Multivariate t-Statistic ....................... 107
5.3.3 Asymptotic Expansions .......................... 109
5.4 Statistics with a Class of Moments .................... 1ll
5.4.1 Large-Sample Expansions ........................ 1ll
5.4.2 High-Dimensional Expansions .................... 117
5.5 Perturbation Method ................................... 120
5.6 Cornish-Fisher Expansions ............................. 125
5.6.1 Expansion Formulas ............................. 125
5.6.2 Validity of Cornish-Fisher Expansions .......... 129
5.7 Transformations for Improved Approximations ........... 132
5.8 Bootstrap Approximations .............................. 135
5.9 High-Dimensional Approximations ....................... 138
5.9.1 Limiting Spectral Distribution ................. 138
5.9.2 Central Limit Theorem .......................... 140
5.9.3 Martingale Limit Theorem ....................... 143
5.9.4 Geometric Representation ....................... 144
Problems ................................................... 145
6 MANOVA Models .............................................. 149
6.1 Multivariate One-Way Analysis of Variance ............. 149
6.2 Multivariate Two-Way Analysis of Variance ............. 152
6.3 MANOVA Tests .......................................... 157
6.3.1 Test Criteria .................................. 157
6.3.2 Large-Sample Approximations .................... 158
6.3.3 Comparison of Powers ........................... 159
6.3.4 High-Dimensional Approximations ................ 161
6.4 Approximations Under Nonnormality ..................... 163
6.4.1 Asymptotic Expansions .......................... 163
6.4.2 Bootstrap Tests ................................ 167
6.5 Distributions of Characteristic Roots ................. 170
6.5.1 Exact Distributions ............................ 170
6.5.2 Large-Sample Case .............................. 172
6.5.3 High-Dimensional Case .......................... 174
6.6 Tests for Dimensionality .............................. 176
6.6.1 Three Test Criteria ............................ 176
6.6.2 Large-Sample and High-Dimensional
Asymptotics .................................... 178
6.7 High-Dimensional Tests ................................ 181
Problems ................................................... 183
7 Multivariate Regression .................................... 187
7.1 Multivariate Linear Regression Model .................. 187
7.2 Statistical Inference ................................. 189
7.3 Selection of Variables ................................ 194
7.3.1 Stepwise Procedure ............................. 194
7.3.2 Cp Criterion ................................... 196
7.3.3 AIC Criterion .................................. 200
7.3.4 Numerical Example .............................. 202
7.4 Principal Component Regression ........................ 203
7.5 Selection of Response Variables ....................... 206
7.6 General Linear Hypotheses and Confidence Intervals .... 209
7.7 Penalized Regression Models ........................... 213
Problems ................................................... 213
8 Classical and High-Dimensional Tests for Covariance
Matrices ................................................... 219
8.1 Specified Covariance Matrix ........................... 219
8.1.1 Likelihood Ratio Test and Moments .............. 219
8.1.2 Asymptotic Expansions .......................... 221
8.1.3 High-Dimensional Tests ......................... 225
8.2 Sphericity ............................................ 227
8.2.1 Likelihood Ratio Tests and Moments ............. 227
8.2.2 Asymptotic Expansions .......................... 228
8.2.3 High-Dimensional Tests ......................... 230
8.3 Intraclass Covariance Structure ....................... 231
8.3.1 Likelihood Ratio Tests and Moments ............. 231
8.3.2 Asymptotic Expansions .......................... 233
8.3.3 Numerical Accuracy ............................. 235
8.4 Test for Independence ................................. 236
8.4.1 Likelihood Ratio Tests and Moments ............. 236
8.4.2 Asymptotic Expansions .......................... 238
8.4.3 High-Dimensional Tests ......................... 239
8.5 Tests for Equality of Covariance Matrices ............. 241
8.5.1 Likelihood Ratio Test and Moments .............. 241
8.5.2 Asymptotic Expansions .......................... 243
8.5.3 High-Dimensional Tests ......................... 244
Problems ................................................... 245
9 Discriminant Analysis ...................................... 249
9.1 Classification Rules for Known Distributions .......... 249
9.2 Sample Classification Rules for Normal Populations .... 256
9.2.1 Two Normal Populations with Σ1 = Σ2 ............ 256
9.2.2 Case of Several Normal Populations ............. 258
9.3 Probability of Misclassifications ..................... 258
9.3.1 W-Rule ......................................... 259
9.3.2 Z-Rule ......................................... 261
9.3.3 High-Dimensional Asymptotic Results ............ 263
9.4 Canonical Discriminant Analysis ....................... 265
9.4.1 Canonical discriminant Method .................. 265
9.4.2 Test for Additional Information ................ 267
9.4.3 Selection of Variables ......................... 270
9.4.4 Estimation of Dimensionality ................... 273
9.5 Regression Approach ................................... 276
9.6 High-Dimensional Approach ............................. 278
9.6.1 Penalized Discriminant Analysis ................ 278
9.6.2 Other Approaches ............................... 278
10 Principal Component Analysis ............................... 283
10.1 Definition of Principal Components .................... 283
10.2 Optimality of Principal Components .................... 286
10.3 Sample Principal Components ........................... 288
10.4 MLEs of the Characteristic Roots and Vectors .......... 291
10.5 Distributions of the Characteristic Roots ............. 292
10.5.1 Exact Distribution ............................. 293
10.5.2 Large-sample Case .............................. 294
10.5.3 High-dimensional Case .......................... 301
10.6 Model Selection Approach for Covariance Structures .... 302
10.6.1 General Approach ............................... 302
10.6.2 Models for Equality of the Smaller Roots ....... 305
10.6.3 Selecting a Subset of Original Variables ....... 306
10.7 Methods Related to Principal Components ............... 308
10.7.1 Fixed-Effect Principal Component Model ......... 308
10.7.2 Random-Effect Principal Components Model ....... 310
Problems ................................................... 311
11 Canonical Correlation Analysis ............................. 317
11.1 Definition of Population Canonical Correlations and
Variables ............................................. 317
11.2 Sample Canonical Correlations ......................... 322
11.3 Distributions of Canonical Correlations ............... 324
11.3.1 Distributional Reduction ....................... 324
11.3.2 Large-Sample Asymptotic Distribuitons .......... 326
11.3.3 High-Dimensional Asymptotic Distributions ...... 327
11.3.4 Fisher's z-Transformation ...................... 333
11.4 Inference for Dimensionality ..................... 335
11.4.1 Test of Dimensionality ......................... 335
11.4.2 Estimation of Dimensionality ................... 337
11.5 Selection of Variables ................................ 338
11.5.1 Test for Redundancy ............................ 338
11.5.2 Selection of Variables ......................... 342
Problems ................................................... 345
12 Growth Curve Analysis ...................................... 349
12.1 Growth Curve Model .................................... 349
12.2 Statistical Inference: One Group ...................... 352
12.2.1 Test for Adequacy .............................. 352
12.2.2 Estimation and Test ............................ 354
12.2.3 Confidence Intervals ........................... 357
12.3 Statistical Methods: Several Groups ................... 359
12.4 Derivation of Statistical Inference ................... 365
12.4.1 General Multivariate Linear Model .............. 365
12.4.2 Estimation ..................................... 366
12.4.3 LR Tests for General Linear Hypotheses ......... 368
12.4.4 Confidence Intervals ........................... 369
12.5 Model Selection ....................................... 370
12.5.1 AIC and CAIC ................................... 370
12.5.2 Derivation of CAIC ............................. 371
12.5.3 Extended Growth Curve Model .................... 373
Problems ................................................... 376
13 Approximation to the Scale-Mixted Distributions ............ 379
13.1 Introduction .......................................... 379
13.1.1 Simple Example: Student's t-Distribution ....... 379
13.1.2 Improving the Approximation .................... 381
13.2 Error Bounds evaluated in sup-Norm .................... 384
13.2.1 General Theory ................................. 384
13.2.2 Scale-Mixed Normal ............................. 388
13.2.3 Scale-Mixed Gamma .............................. 390
13.3 Error Bounds evaluated in L1-Norm ..................... 395
13.3.1 Some Basic Results ............................. 395
13.3.2 Scale-Mixed Normal Density ..................... 397
13.3.3 Scale-Mixed Gamma Density ...................... 399
13.3.4 Scale-Mixed Chi-square Density ................. 402
13.4 Multivariate Scale Mixtures ........................... 404
13.4.1 General Theory ................................. 404
13.4.2 Normal Case .................................... 410
13.4.3 Gamma Case ..................................... 415
Problems ................................................... 418
14 Approximation to Some Related Distributions ................ 423
14.1 Location and Scale Mixtures ........................... 423
14.2 Maximum of Multivariate Variables ..................... 426
14.2.1 Distribution of the Maximum Component of
a Multi-variate Variable ....................... 426
14.2.2 Multivariate t-Distribution .................... 427
14.2.3 Multivariate F-Distribution .................... 429
14.3 Scale Mixtures of the F-Distribution .................. 430
14.4 Nonuniform Error Bounds ............................... 433
14.5 Method of Characteristic Functions .................... 436
Problems ................................................... 439
15 Error Bounds for Approximations of Multivariate Tests ...... 441
15.1 Multivariate Scale Mixture and MANOVA Tests ........... 441
15.2 Function of a Multivariate Scale Mixture .............. 443
15.3 Hotelling's T20, Statistic ............................. 445
15.4 Wilk's Lambda Distribution ............................ 448
15.4.1 Univariate Case ................................ 448
15.4.2 Multivariate Case .............................. 456
Problems ................................................... 465
16 Error Bounds for Approximations to Some Other Statistics ... 467
16.1 Linear Discriminant Function .......................... 467
16.1.1 Representation as a Location and Scale
Mixture ........................................ 467
16.1.2 Large-Sample Approximations .................... 472
16.1.3 High-Dimensional Approximations ................ 474
16.1.4 Some Related Topics ............................ 476
16.2 Profile Analysis ...................................... 479
16.2.1 Parallelism Model and MLE ...................... 479
16.2.2 Distributions of  ............................. 481
16.2.3 Confidence Interval for γ ...................... 486
16.3 Estimators in the Growth Curve Model .................. 487
16.3.1 Error Bounds ................................... 487
16.3.2 Distribution of the Bilinear Form .............. 488
16.4 Generalized Least Squares Estimators .................. 490
Problems ................................................... 492
Appendix ...................................................... 495
A.1 Some Results on Matrices .............................. 495
A.1.1 Determinants and Inverse Matrices .............. 495
A.1.2 Characteristic Roots and Vectors ............... 496
A.1.3 Matrix Factorizations .......................... 497
A.1.4 Idempotent Matrices ............................ 500
A.2 Inequalities and Max-Min Problems ..................... 502
A.3 Jacobians of Transformations .......................... 508
Bibliography ............................................... 513
Index ...................................................... 527
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