Multivariate Statistical Analysis: Revised And Expanded

by Giri; Narayan C.

• ISBN13: 9780824747138
• ISBN10: 0824747135
• Edition: 2nd
• Format: Hardcover
• Copyright: 2003-11-14
• Publisher: CRC Press

Summary

This Second Edition adds new topics in properties and characterization of symmetric distribution, elliptically symmetric multivariate distributions, singular symmetric distributions, estimation of covariance matrices, tests of mean against one-sided alternatives, and more. Draws on multivariate data from biometry, agriculture, biomedical sciences, economics, filtering and stochastic control, stock market data analysis, and random signal processing.

Author Biography

Narayan C. Giri is Professor of Mathematics and Statistics, University of Montreal, Quebec, Canada.

Table of Contents

Preface to the Second Edition	p. v
Preface to the First Edition	p. vii
Vector and Matrix Algebra	p. 1
Introduction	p. 1
Vectors	p. 1
Matrices	p. 4
Rank and Trace of a Matrix	p. 7
Quadratic Forms and Positive Definite Matrix	p. 7
Characteristic Roots and Vectors	p. 8
Partitioned Matrix	p. 16
Some Special Theorems on Matrix Derivatives	p. 21
Complex Matrices	p. 24
Exercises	p. 25
References	p. 27
Groups, Jacobian of Some Transformations, Functions and Spaces	p. 29
Introduction	p. 29
Groups	p. 29
Some Examples of Groups	p. 30
Quotient Group, Homomorphism, Isomorphism	p. 31
Jacobian of Some Transformations	p. 33
Functions and Spaces	p. 38
References	p. 39
Multivariate distributions and Invariance	p. 41
Introduction	p. 41
Multivariate Distributions	p. 41
Invariance in Statistical Testing of Hypotheses	p. 44
Almost Invariance and Invariance	p. 49
Sufficiency and Invariance	p. 55
Unbiasedness and Invariance	p. 56
Invariance and Optimum Tests	p. 57
Most Stringent Tests and Invariance	p. 58
Locally Best and Uniformly Most Powerful Invariant Tests	p. 58
Ratio of Distributions of Maximal Invariant, Stein's Theorem	p. 59
Derivation of Locally Best Invariant Tests (LBI)	p. 61
Exercises	p. 63
References	p. 65
Properties of Multivariate Distributions	p. 69
Introduction	p. 69
Multivariate Normal Distribution (Classical Approach)	p. 70
Complex Multivariate Normal Distribution	p. 84
Symmetric Distribution: Its Properties and Characterizations	p. 91
Concentration Ellipsoid and Axes (Multivariate Normal)	p. 110
Regression, Multiple and Partial Correlation	p. 112
Cumulants and Kurtosis	p. 118
The Redundancy Index	p. 120
Exercises	p. 120
References	p. 127
Estimators of Parameters and Their Functions	p. 131
Introduction	p. 131
Maximum Likelihood Estimators of [mu], [Sigma] in N[subscript p]([mu], [Sigma])	p. 132
Classical Properties of Maximum Likelihood Estimators	p. 141
Bayes, Minimax, and Admissible Characters	p. 151
Equivariant Estimation Under Curved Models	p. 184
Exercises	p. 202
References	p. 206
Basic Multivariate Sampling Distributions	p. 211
Introduction	p. 211
Noncentral Chi-Square, Student's t-, F-Distributions	p. 211
Distribution of Quadratic Forms	p. 213
The Wishart Distribution	p. 218
Properties of the Wishart Distribution	p. 224
The Noncentral Wishart Distribution	p. 231
Generalized Variance	p. 232
Distribution of the Bartlett Decomposition (Rectangular Coordinates)	p. 233
Distribution of Hotelling's T[superscript 2]	p. 234
Multiple and Partial Correlation Coefficients	p. 241
Distribution of Multiple Partial Correlation Coefficients	p. 245
Basic Distributions in Multivariate Complex Normal	p. 248
Basic Distributions in Symmetrical Distributions	p. 250
Exercises	p. 258
References	p. 264
Tests of Hypotheses of Mean Vectors	p. 269
Introduction	p. 269
Tests: Known Covariances	p. 270
Tests: Unknown Covariances	p. 272
Tests of Subvectors of [mu] in Multivariate Normal	p. 299
Tests of Mean Vector in Complex Normal	p. 307
Tests of Means in Symmetric Distributions	p. 309
Exercises	p. 317
References	p. 320
Tests Concerning Covariance Matrices and Mean Vectors	p. 325
Introduction	p. 325
Hypothesis: A Covariance Matrix Is Unknown	p. 326
The Sphericity Test	p. 337
Tests of Independence and the R[superscript 2]-Test	p. 342
Admissibility of the Test of Independence and the R[superscript 2]-Test	p. 349
Minimax Character of the R[superscript 2]-Test	p. 353
Multivariate General Linear Hypothesis	p. 369
Equality of Several Covariance Matrices	p. 389
Complex Analog of R[superscript 2]-Test	p. 406
Tests of Scale Matrices in E[subscript p]([mu], [Sigma])	p. 407
Tests with Missing Data	p. 412
Exercises	p. 423
References	p. 427
Discriminant Analysis	p. 435
Introduction	p. 435
Examples	p. 437
Formulation of the Problem of Discriminant Analysis	p. 438
Classification into One of Two Multivariate Normals	p. 444
Classification into More than Two Multivariate Normals	p. 468
Concluding Remarks	p. 473
Discriminant Analysis and Cluster Analysis	p. 473
Exercises	p. 474
References	p. 477
Principal Components	p. 483
Introduction	p. 483
Principal Components	p. 483
Population Principal Components	p. 485
Sample Principal Components	p. 490
Example	p. 492
Distribution of Characteristic Roots	p. 495
Testing in Principal Components	p. 498
Exercises	p. 501
References	p. 502
Canonical Correlations	p. 505
Introduction	p. 505
Population Canonical Correlations	p. 506
Sample Canonical Correlations	p. 510
Tests of Hypotheses	p. 511
Exercises	p. 514
References	p. 515
Factor Analysis	p. 517
Introduction	p. 517
Orthogonal Factor Model	p. 518
Oblique Factor Model	p. 519
Estimation of Factor Loadings	p. 519
Tests of Hypothesis in Factor Models	p. 524
Time Series	p. 525
Exercises	p. 526
References	p. 526
Bibliography of Related Recent Publications	p. 529
Tables for the Chi-Square Adjustment Factor	p. 531
Publications of the Author	p. 543
Author Index	p. 551
Subject Index	p. 555
Table of Contents provided by Ingram. All Rights Reserved.