xiiContentsSAMPLESFROMAMULTIVARIATENORMALDISTRIBUTION,3.ANDTHEWISHARTANDMULTIVARIATEBETA79DISTRIBUTIONS3.1.SamplesFromaMultivariateNormal Distribution andMaximumLikelihood Estimation of theParameters,793.2.The WishartDistribution,853.2.1.The Wishart Density Function, 853.2.2.CharacteristicFunction,Moments,and AsynptoticDistribution,873.2.3.Some Properties of the Wishart Distribution,913.2.4.Bartlett's Decomposition and the ieneralized Variance,993.2.5.The Latent Roots of a Wishart Matrix, 1033.3.The MultivariateBeta Distribution, 108Problems,1124.SOMERESULTSCONCERNINGDECISION-THEORETICESTIMATIONOFTHEPARAMETERSOFAMULTIVARIATE121NORMALDISTRIBUTION4.1.Introduction,1214.2.EstimalionoftheMean,1224.3.Estimationof theCovarianceMatrix,1284.4.Estimation of the Precision Matrix,136Problems,1411445.CORRELATIONCOEFFICIENTS5.1.OrdinaryCorrelationCoefficients,1445.1.1.Introduction,1445.1.2.Joint and MarginalDistributions of SampleCorrelationCoefficients in the Case of Independence, 1455.1.3.The Non-null Distribution of a Sample CorrelationCoefficient in the Case of Normality,15!5.1.4.AsymptoticDistributionofaSampleCorreiationCoefficientFrom an EllipticalDistribution,1575.1.5,Testing Hypotheses about Population CorrelationCoefficients,1605.2.The Multiple Correlation Coefficient, 1645.2.1.Introduction,1645.2.2.Distributionof the SampleMultiple CorrelationCoefficicntin the Case of Independence, 167
3. SAMPLES FROM A MULTIVARIATE NORMAL I)ISTRIBUTION, AND THE WISHAIlT AND MULTIVARIATE BETA DISTRIDUTIONS 79 3. I. 3.2. The Wishart Distribution, 85 Samples From a Multivariate Normal Distribution and Maximum Likelihood Estimation of the Parameters, 79 3.2.1. The Wishart Density Function, 85 3.2.2. Characteristic Function, Moments, and Asymptotic Distribution, 57 3.2.3. Some Properties of the Wishart Distribution, 91 3.2,4. Bartlett’s Dccomposition and the Generalized Variance, 99 3.2.5. The Latent Roots of a Wishart Matrix, 103 3.3. The Multivariate Beta Distribution, 108 Problems, 112 4. SOME RESULTS CONCERNING DECISION-THEORETIC ESTIMATION OF THE PARAMETERS OF A MULTIVARIATE NORMAL DISTRIBUTION 121 4.1. Introduction, 12 I 4.2. Estimation of the Mean, 122 4.3. Estimation of the Covariance Matrix, 128 4.4. Estimation of the Precision Matrix, 136 Problems, 14 I 5. CORRELATION COEFFICIENTS 5. I. Ordinary Correlation Coefficients, 144 5.1.1. Introduction, 144 5. I .2. Joint and Marginal Distributions of Sample Correlation Coefficients in the Case of Independence, 145 5.1.3. The Non-null Distribution of a Sample Correlation Coefficient in the Case of Normality, I51 5.1.4. Asymptotic Distribution of a Sample Correlation Coefficient From an Elliptical Distribution, 157 5. I .5. Testing Hypothcses about I’opulation Correlation Coefficients, 160 5.2. The Multiple Correlation Coefficient, 164 5.2. I. Introduction, 164 5.2.2. Distribution of the Sample Multiple Correlation Coefficicnt in the Case of Independcncc, 167 144
xiliContents5.2.3.The Non-null Distribution of a Sample Multiple CorrelationCoefficient in the Case of Normality,17t5.2.4.Asymptotic Distributions of a Sample Multiple CorrelationCoefficientfromanElliptical Distribution,1795.2.5.Testing Hypotheses about a Population MultipleCorrelationCoefficient,1855.3.Partial CorrelationCoefficients,187Problems,1891966.INVARIANTTESTSANDSOMEAPPLICATIONS6.1.Invariance and Invariant Tests,1966.2.The MultipleCorrelation Coefficientand Invariance,2066.3.Hotelling's T? Statistic and Invariance,211Problems,219ZONALPOLYNOMIALSANDSOMEFUNCTIONSOF7.225MATRIXARGUMENT7.1.Introduction,2257.2.2ZonalPolynomials,2277.2.1.Definition and Construction,2277.2.2.AFundamentalProperty,2397.2.3.SomeBasicIntegrals,2467.3.Hypergeometric Functionsof MatrixArgument,2587.4.Some Resulls on Special HypergeometricFunctions,2627.5..Partial Differential EquationsforHypergcometricFunctions,2667.6.Generalized LaguerrePolynomials,281Problems,286SOMESTANDARDTESTSONCOVARIANCEMATRICESAND8.291MEANVECTORS8.1.Introduction,2918.2.Testing Equality of r Covariance Matrices,2918.2.1.TheLikelihoodRatioStatistic and Invariance,2918.2.2.Unbiasedness and theModified Likelihood RatioTest,2968.2.3.Central Momentsofthe ModifiedLikelihood RatioStatistic,3018.2.4.The Asymptotic Null Distribution of the ModifiedLikelihood Ratio Statistic,303
Conrenrs xiii 5.2.3. The Non-null Distribution of a Sample Multipfe Correlation Coefficient in the Case of Normality, 171 5.2.4. Asymptotic Distributions of a Sample Multiple Correlation Coefficient from an Elliptical Distribution, I79 5.2.5. Testing Hypotheses about a Population Multiple Correlation Coefficient, 185 5.3. Partial Correlation Coefficients, 187 Problems, I89 6. INVARIANT TESTS AND SOME APPLICATIONS 196 6. I. Invariance and Invariant Tests, 196 6.2. The Multiple Correlation Coefficient and Invariance, 206 6.3. Hotelling’s T2 Statistic and Invariance, 21 I Problems, 2 I9 7. ZONAL POLYNOMIALS AND SOME FUNCTIONS OF MATRIX ARGUMENT 225 7. I. Introduction, 225 7.2. Zonal Polynomials, 227 7.2. I. 7.2.2. A Fundamental Property, 239 7.2.3. Some Basic Integrals, 246 7.3. Hypergeometric Functions of Matrix Argument, 258 7.4. Some Results on Special Hypergeometric Functions, 262 7.5. Partial Differential Equations for Hypergeometric Functions, 266 7.6. Generalized Laguerre Polynomials, 28 I Problems, 286 Definition and Construction, 227 8. SOME STANDARD TESTS ON COVARIANCE MATRICES AND MEAN VECTORS 29 I 8.1. Introduction, 291 8.2. Testing Equality of r Covariance Matrices, 291 8.2.1. The Likelihood Ratio Statistic and Invariance, 291 8.2.2. Unbiasedness and the Modified Likelihood Ratio Test, 296 8.2.3. Central Moments of the Modified Likelihood Ratio Statistic, 301 8.2.4. The Asymptotic Null Distribution of the Modified Likelihood Ratio Statistic, 303
xivContents8.2.5.Noncentral Moments oftheModified LikelihoodRatioStatistic when r=2, 3118.2.6.Asymptotic Non-null Distributions of the ModifiedLikelihoodRatio Statistic when r=2,3168.2.7.The Asymptotic Null Distribution of the ModifiedLikelihood Ratio Statistic for Elliptical Samples,3298.2.8.Other Test Statistics, 3318.3.The SphericityTest, 3338.3.1.TheLikclihood Ratio Statistic,InvarianceandUnbiascdness,3338.3.2.Momcnts of theLikelihood RatioStatistic,3398.3.3.The Asynptotic Null Distribution of the Likelihood RatioStatistic,3438.3.4.Asymptotic Non-null Distributions of the Likelihood RatioStatistic,3448.3.5.The Asymptotic Null Distribution of the Likelihood RatioStatisticforanEllipticalSample,3518.3.6.OtherTest Statistics,3538.4.Testing That a Covariance Matrix Equals a Specified Matrix,3538.4.1.TheLikelihoodRatioTest andInvariance,3538.4.2.Unbiascdnessand theModifiedLikelihoodRatioTest,3568.4.3.Momentsof theModificdLikelihoodRatioStatistic,3588.4.4.The Asymptotic Null Distribution of the ModificdLikelihoodRatioStatistic,3598.4.5.Asymptotic Non-null Distributions of the ModifiedLikelihoodRatioStatistic,3628.4.6.The Asymptotic Null Distribution of the ModifiedLikelihood Ratio Statistic for an Elliptical Sample,3648.4.7.OtherTest Statistics,3658.5.Testing Specified Values for the MeanVector and CovarianceMatrix,3668.5.1.The Likelihood Ratio Test, 3668.5.2.MomentsoftheLikelihoodRatioStatistic,3698.5.3.The Asymptotic Null Distribution of the Likelihood RatioStatistic,3708.5.4Asymptotic Non-null Distributions of the Likelihood RatioStatistic,373Problems, 3763809.PRINCIPALCOMPONENTSANDRELATEDTOPICS9.1.Introduction,3809.2.Population Principal Components,3819.3.SamplePrincipalComponents,384
8.2.5. Noncentral Moments of the Modified Likelihood Ratio Statistic when r = 2, 3 I 1 8.2.6. Asymptotic Non-null Distributions of the Modificd Likelihood Ratio Statistic when r = 2, 3 16 8.2.7. The Asymptotic Null Distribution of the Modificd Likelihood Ratio Statistic for Elliptical Samples, 329 8.2.8. Other Test Statistics, 33 I 8.3. I. The Likelihood Ratio Statistic; Invariance and Unbiasedness, 333 8.3.2. Momcnts of the Likelihood Ratio Statistic. 339 8.3.3. The Asymptotic Null Distribution of tlic Likelihood Ratio Statistic, 343 8.3.4. Asymptotic Non-null Distributions of the Likelihood Ratio Statistic, 344 8.3.5. The Asymptotic Null Distribution of the Likelihood Ratio Statistic for an Elliptical Sample, 351 8.3.6. Other Test Statistics, 353 Testing That a Covariance Matrix Equals a Specified Matrix, 353 8.4. I. The Likelihood Ratio Test and Invariance, 353 8.4.3. Unbiascdness and the Modified Likelihood Ratio Test, 356 8.4.3. Mornents of the Modified Likelihood Ratio Statistic, 358 8.4.4. The Asymptotic Null Distribution of the Modificd Likelihood Ratio Statistic, 359 8.4.5. Asymptotic Non-null Distributions of the Modified Likelihood Ratio Statistic, 362 8.4.6. The Asymptotic Null Distribution of the Modified Likelihood Ratio Statistic for an Elliptical Sample, 364 8.4.7. Other Test Statistics, 365 8.5. Testing Specified Values for the Mean Vector and Covariance Matrix, 366 8.5.1. The Likelihood Ratio Test, 366 8.5.2. Moments of the Likelihood Ratio Statistic, 369 8.5.3. The Asymptotic Null Distribution of the Likelihood Ratio Statistic, 370 8.5.4. Asymptotic Non-null Distributions of the Likelihood Ratio Statistic, 373 8.3. The Sphericity Test, 333 8.4. Problems, 376 9. PRINCIPAL COMPONENTS AND RELATED TOPICS 380 9. I. Introduction, 380 9.2. 9.3. Sample Principal Components, 384 Population Principal Components, 38 I
xVContents9.4.TheJointDistributionof the LatentRoots of aSampleCovarianceMatrix,3889.5.Asymptotic Distributions of the Latent Roots of a SampleCovariance Matrix, 3909.6.Some InferenceProblems inPrincipal Components,4059.7.Distributions of the Extreme Latent Roots of a Sample CovarianceMatrix,420Problems, 426429IO.THEMULTIVARIATELINEARMODELIntroduction,42910.110.2.AGeneralTestingProblem:CanonicalForm,Invariance,andtheLikelihoodRatioTest,43210.3.The Noncentral Wishart Distribution,44110.4.Joint Distributions of LatentRoots in MANOVA,44910.5.Distributional Results for the Likelihood Ratio Statistic,45510.5.1.Moments,45510.5.2.NullDistribution,45710.5.3.TheAsymptoticNullDistribution,45810.5.4.AsymptoticNon-null Distributions,46010.6OtherTest Statistics,46510.6.1.Introduction,46510.6.2.The T2 Statistic, 46610.6.3.TheVStatistic,47910.6.4.The Largest Root, 48110.6.5.PowerComparisons,48410.7.The Single Classification Model,48510.7.1.Introduction,48510.7.2.MultipleDiscriminantAnalysis,48810.7.3.Asymptotic Distributions of Latent Roots in MANOVA,49210.7.4.Determining the Number of Useful DiscriminantFunctions,49910.7.5.Discrimination Between TwoGroups,50410.8.TestingEqualityofp Normal Populations,50710.8.1.The Likelihood Ratio Statistic and Moments,50710.8.2.The Asymptotic Null Distribution of the LiketihoodRatio Statistic, 51210.8.3.An Asymptotic Non-null Distribution of the LikelihoodRatio Statistic,513Problems,517
Corifetirs xv 9.4. The Joint Distribution of the Latent Roots of a Sample Covariance Matrix, 388 9.5. Asymptotic Distributions of the Latent Roots of a Sample Covariance Matrix, 390 9.6. Some Inference Problems in Principal Components, 405 9.7. Distributions of the Extreme Latent Roots of a Sample Covariance Matrix, 420 Problems, 426 10. THE MULTIVARIATE LINEAR MODEL 10. I. Introduction, 429 10.2. A General Testing Problem: Canonical Form, Invariance, and the Likelihood Ratio Test, 432 10.3. The Noncentral Wishart Distribution, 441 10.4. Joint Distributions of Latent Roots in MANOVA, 449 10.5. Distributional Results for the Likelihood Ratio Statistic, 455 10.5.1. Moments, 455 10.5.2. Null Distribution, 457 10.5.3. The Asymptotic Null Distribution, 458 10.5.4. Asymptotic Non-null Distributions, 460 10.6. I. Introduction, 465 10.6.2. The T: Statistic, 466 10.6.3. The V Statistic, 479 10.6.4. The Largest Root, 481 10.6.5. Power Comparisons, 484 10.7. The Single Classification Model, 485 10.7. I. Introduction, 485 10.7.2. Multiple Discriminant Analysis, 488 10.7.3. Asymptotic Distributions of Latent Roots in MANOVA, 492 10.7.4. Determining the Number of Useful Discriminant Functions, 499 10.7.5. Discrimination Between Two Groups, 504 Testing Equality of p Normal Populations, 507 10.8.1. The Likelihood Ratio Statistic and Moments, 507 10.8.2. The Asymptotic Null Distribution of the Liketihood Ratio Statistic, 5 12 10.8.3. An Asymptotic Non-null Distribution of the Likelihood Ratio Statistic, 5 13 Problems, 5 17 10.6 Other Test Statistics, 465 10.8. 429
xviContents11.TESTINGINDEPENDENCEBETWEENKSETSOFVARIABLES526ANDCANONICALCORRELATIONANALYSIS11.1.Introduction,52611.2.TestingIndependenceof kSetsof Variables,52611.2.1.The Likelihood Ratio Statistic and Invariance, 52611.2.2.CcntralMomentsoftheLikelihoodRatioStatistic,53211.2.3.TheNull Distribution of theLikelihood RatioStatistic,53311.2.4.The Asymplotic Null Distribution of theLikelihoodRatioStatistic,53411.2.5.Noncentral Moments of the Likelihood Ratio Statisticwhenk=2.53611.2.6AsymptoticNon-null Distributions of the LikelihoodRatioStatisticwhenk=2,54211.2.7.TheAsymptoticNullDistributionoftheLikelihoodRatio Statistic for Elliptical Samples,54611.2.8.Other Test Statistics, 54811.3.Canonical Correlation Analysis,54811.3.1.Introduction,54811.3.2.Population Canonical Correlation Coefficicnts andCanonical Variables,54911.3.3.SampleCanonical Correlation Coefficients andCanonical Variables, 55511.3.4.Distributions of the Sample Canonical CorrelationCoefficients,55711.3.5.Asymptotic Distributions of the Sample CanonicalCorrelation Cocfficients,56211.3.6.Determining the Number of Useful CanonicalVariables,567Problems,569572APPENDIXSOMEMATRIXTHEORYAl.Introduction,572A2.Definitions,572A3,Determinants,575A4.Minors and Cofactors,579A5.Inverse of a Matrix, 579A6.Rankof aMatrix,582A7.LatentRoots andLatcntVectors,582A8.PositiveDefiniteMatrices,585A9.SomeMatrixFactorizations,S86650BIBLIOGRAPHYINDEX663
xvi Conretiis I I. TESTING INDEPENDENCE BETWEEN k SETS OF VARIABLES AND CANONICAL CORRELATION ANALYSIS 526 11.1. Introduction, 526 1 1.2. Testing Independence of k Sets of Variables, 526 I I .2. I. The Likelihood Ratio Statistic and Invariance, 526 11.2.2. Central Moments of the Likelihood Ratio Statistic, 532 11.2.3. The Null Distribution of the Likelihood Katio Statistic, 533 11.2.4. The Asymptotic Null Distribution of the Likelihood Ratio Statistic, 534 I 1.2.5. Nonceritral Moments of the Likelihood Ratio Statistic when k = 2, 536 I I .2.6 Asymptotic Non-null Distributions of the Likelihood Ratio Statistic when k = 2, 542 11.2.7. The Asymptotic Null Distribution of the Likelihood Ratio Statistic for Elliptical Samples, 546 11.2.8. Other Test Statistics, 548 1 I .3. Canonical Correlation Analysis, 548 11.3.1. Introduction, 548 I I .3.2. Population Canonical Correlation Coefficicnts and Canonical Variables, 549 I 1.3.3. Sample Canonical Correlation Coefficients and Canonical Variables, 555 11.3.4. Distributions of the Sample Canonical Correlation Coefficients, 557 I 1-33. Asymptotic Distributions of the Sample Canonical Correlation Coefficients, 562 I 1.3.6. Determining the Number of Useful Canonical Variables, 567 Problems. 569 APPENDIX. SOME MATRIX THEORY Al. A2. A3. A4. AS. A6. A7. A8. A9. BIBLIOGRAPHY INDEX 572 Introduction, 572 Definitions, 572 Determinants, 575 Minors and Cofactors, 579 Inverse of a Matrix, 579 Rank of a Matrix, 582 Latent Roots and Latent Vectors, 582 Positive Definite Matrices, 585 Some Matrix Factorizations, 586 650 663