Multivariate Statistics: A Vector Space Approach

Multivariate Statistics: A Vector Space Approach......Page 1
Contents......Page 4
Preface......Page 6
Notation......Page 9
C H A P T E R 1
Vector Space Theory......Page 10
VECTOR SPACES......Page 11
LINEAR TRANSFORMATIONS......Page 15
INNER PRODUCT SPACES......Page 22
THE CAUCHY-SCHWARZ INEQUALITY......Page 34
THE SPACE C(V, W)......Page 38
DETERMINANTS AND EIGENVALUES......Page 47
THE SPECTRAL THEOREM......Page 58
PROBLEMS......Page 72
NOTES AND REFERENCES......Page 78
CHAPTER 2: Random Vectors......Page 0
RANDOM VECTORS......Page 79
INDEPENDENCE OF RANDOM VECTORS......Page 85
SPECIAL COVARIANCE STRUCTURES......Page 90
PROBLEMS......Page 107
NOTES AND REFERENCES......Page 111
C H A P T E R 3
The Normal Distribution
on a Vector Space......Page 112
THE NORMAL DISTRIBUTION......Page 113
QUADRATIC FORMS......Page 118
INDEPENDENCE OF QUADRATIC FORMS......Page 122
CONDITIONAL DISTRIBUTIONS......Page 125
THE DENSITY OF THE NORMAL DISTRIBUTION......Page 129
PROBLEMS......Page 136
NOTES AND REFERENCES......Page 140
THE CLASSICAL LINEAR MODEL......Page 141
MORE ABOUT THE GAUSS-MARKOV THEOREM......Page 149
GENERALIZED LINEAR MODELS......Page 155
PROBLEMS......Page 163
NOTES AND REFERENCES......Page 166
MATRIX FACTORIZATIONS......Page 168
JACOBIANS......Page 175
PROBLEMS......Page 189
NOTES AND REFERENCES......Page 192
CHAPTER 6: Topological Groups
and Invariant Measures......Page 193
GROUPS......Page 194
INVARIANT MEASURES AND INTEGRALS......Page 203
INVARIANT MEASURES ON QUOTIENT SPACES......Page 216
TRANSFORMATIONS AND FACTORIZATIONS OF MEASURES......Page 227
PROBLEMS......Page 237
NOTES AND REFERENCES......Page 241
LEFT 0n INVARIANT DISTRIBUTIONS ON n X p MATRICES......Page 242
GROUPS ACTING ON SETS......Page 250
INVARIANT PROBABILITY MODELS......Page 260
THE INVARIANCE OF LIKELIHOOD METHODS......Page 267
DISTRIBUTION THEORY AND INVARIANCE......Page 276
INDEPENDENCE AND INVARIANCE......Page 293
PROBLEMS......Page 305
NOTES AND REFERENCES......Page 308
BASIC PROPERTIES......Page 311
PARTITIONING A WISHART MATRIX......Page 318
THE NONCENTRAL WISHART DISTRIBUTION......Page 325
DISTRIBUTIONS RELATED TO LIKELIHOOD RATIO TESTS......Page 327
PROBLEMS......Page 338
NOTES AND REFERENCES......Page 341
CHAPTER 9: Inference for Means
in Multivariate
Linear Models......Page 343
THE MANOVA MODEL......Page 345
MANOVA PROBLEMS WITH BLOCK DIAGONAL COVARIANCE STRUCTURE......Page 359
INTRACLASS COVARIANCE STRUCTURE......Page 364
SYMMETRY MODELS: AN EXAMPLE......Page 370
COMPLEX COVARIANCE STRUCTURES......Page 379
ADDITIONAL EXAMPLES OF LINEAR MODELS......Page 390
PROBLEMS......Page 406
NOTES AND REFERENCES......Page 410
POPULATION CANONICAL CORRELATION COEFFICIENTS......Page 412
SAMPLE CANONICAL CORRELATIONS......Page 428
SOME DISTRIBUTION THEORY......Page 436
TESTING FOR INDEPENDENCE......Page 452
MULTIVARIATE REGRESSION......Page 460
PROBLEMS......Page 465
NOTES AND REFERENCES......Page 472
Appendix......Page 473
Comments on Selected Problems......Page 479
CHAPTER 2......Page 482
CHAPTER 3......Page 484
CHAPTER 4......Page 485
CHAPTER 5......Page 489
CHAPTER 6......Page 491
CHAPTER 7......Page 495
CHAPTER 8......Page 498
CHAPTER 9......Page 501
CHAPTER 10......Page 503
BIBLIOGRAPHY......Page 510
Index......Page 514