xiliContents988Tests on mean and variance988.1Introduction988.2Hotelling-T28.3104Simultaneous confidence intervals on means1048.3.1Linearhypotheses1078.3.2Nonlinearhypotheses1098.4Multiplecorrelation1148.4.1Asymptoticmoments8.5116Partial correlation8.6117Test of sphericity8.7121Test of equality of variances8.8124Asymptotic distributions of eigenvalues1248.8.1The one-sampleproblem1328.8.2Thetwo-sampleproblem1338.8.3Thecaseof multipleeigenvalues8.9137Problems144Multivariateregression99.1144Introduction9.2145Estimation9.3148Thegeneral linear hypothesis1489.3.1Canonicalform1509.3.2LRTforthecanonicalproblem1519.3.3Invariant tests1549.4Random design matrix X9.5156Predictions9.6158One-way classification9.7159Problems16110Principalcomponents16110.1Introduction10.2Definition and basicproperties16216310.3Bestapproximatingsubspace16410.4SampleprincipalcomponentsfromS16610.5Sampleprincipal components fromR16910.6Atestformultivariatenormality10.7Problems17217411 Canonical correlations17411.1Introduction17511.2Definition and basic properties17711.3Testsofindependence18111.4Properties of U distributions18411.4.1 Q-Qplotof squaredradii
Contents xiii 8 Tests on mean and variance 98 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.2 Hotelling-T2 . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3 Simultaneous confidence intervals on means . . . . . . . 104 8.3.1 Linear hypotheses . . . . . . . . . . . . . . . . . . 104 8.3.2 Nonlinear hypotheses . . . . . . . . . . . . . . . . 107 8.4 Multiple correlation . . . . . . . . . . . . . . . . . . . . . 109 8.4.1 Asymptotic moments . . . . . . . . . . . . . . . . 114 8.5 Partial correlation . . . . . . . . . . . . . . . . . . . . . . 116 8.6 Test of sphericity . . . . . . . . . . . . . . . . . . . . . . 117 8.7 Test of equality of variances . . . . . . . . . . . . . . . . 121 8.8 Asymptotic distributions of eigenvalues . . . . . . . . . . 124 8.8.1 The one-sample problem . . . . . . . . . . . . . . 124 8.8.2 The two-sample problem . . . . . . . . . . . . . . 132 8.8.3 The case of multiple eigenvalues . . . . . . . . . . 133 8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9 Multivariate regression 144 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.3 The general linear hypothesis . . . . . . . . . . . . . . . 148 9.3.1 Canonical form . . . . . . . . . . . . . . . . . . . 148 9.3.2 LRT for the canonical problem . . . . . . . . . . 150 9.3.3 Invariant tests . . . . . . . . . . . . . . . . . . . . 151 9.4 Random design matrix X . . . . . . . . . . . . . . . . . . 154 9.5 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9.6 One-way classification . . . . . . . . . . . . . . . . . . . . 158 9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10 Principal components 161 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 161 10.2 Definition and basic properties . . . . . . . . . . . . . . . 162 10.3 Best approximating subspace . . . . . . . . . . . . . . . . 163 10.4 Sample principal components from S . . . . . . . . . . . 164 10.5 Sample principal components from R . . . . . . . . . . . 166 10.6 A test for multivariate normality . . . . . . . . . . . . . 169 10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 11 Canonical correlations 174 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.2 Definition and basic properties . . . . . . . . . . . . . . . 175 11.3 Tests of independence . . . . . . . . . . . . . . . . . . . . 177 11.4 Properties of U distributions . . . . . . . . . . . . . . . . 181 11.4.1 Q-Q plot of squared radii . . . . . . . . . . . . . . 184
Contentsxiv18911.5Asymptoticdistributions11.6Problems19019512Asymptotic expansions19512.1Introduction12.2195General expansions20012.3Examples12.4205Problem20613Robustness20613.1Introduction13.2207Elliptical distributions21313.3Maximumlikelihoodestimates21313.3.1NormalMLE21313.3.2EllipticalMLE22213.4Robust estimates22213.4.1 M estimate.22413.4.2Sestimate13.4.3 Robust Hotelling-T222622713.5Robusttests on scalematrices22813.5.1Adjusted likelihood ratio tests23313.5.2Weighted Nagao's test for a given variance23613.5.3Relativeefficiencyof adjusted LRT23813.6Problems.24314Bootstrap confidenceregions and tests24314.1Confidenceregions and testsforthe mean24614.2Confidence regions for the variance24914.3Tests on the variance25214.4Problem253A Inversion formulas256B Multivariate cumulants256B.1Definition andproperties259B.2Application toasymptotic distributionsB.3259Problems..261C S-plus functionsReferences263277Author IndexSubject Index281
xiv Contents 11.5 Asymptotic distributions . . . . . . . . . . . . . . . . . . 189 11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 12 Asymptotic expansions 195 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 195 12.2 General expansions . . . . . . . . . . . . . . . . . . . . . 195 12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 12.4 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 13 Robustness 206 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 206 13.2 Elliptical distributions . . . . . . . . . . . . . . . . . . . 207 13.3 Maximum likelihood estimates . . . . . . . . . . . . . . . 213 13.3.1 Normal MLE . . . . . . . . . . . . . . . . . . . . 213 13.3.2 Elliptical MLE . . . . . . . . . . . . . . . . . . . 213 13.4 Robust estimates . . . . . . . . . . . . . . . . . . . . . . 222 13.4.1 M estimate . . . . . . . . . . . . . . . . . . . . . . 222 13.4.2 S estimate . . . . . . . . . . . . . . . . . . . . . . 224 13.4.3 Robust Hotelling-T2 . . . . . . . . . . . . . . . . 226 13.5 Robust tests on scale matrices . . . . . . . . . . . . . . . 227 13.5.1 Adjusted likelihood ratio tests . . . . . . . . . . . 228 13.5.2 Weighted Nagao’s test for a given variance . . . . 233 13.5.3 Relative efficiency of adjusted LRT . . . . . . . . 236 13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14 Bootstrap confidence regions and tests 243 14.1 Confidence regions and tests for the mean . . . . . . . . 243 14.2 Confidence regions for the variance . . . . . . . . . . . . 246 14.3 Tests on the variance . . . . . . . . . . . . . . . . . . . . 249 14.4 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 A Inversion formulas 253 B Multivariate cumulants 256 B.1 Definition and properties . . . . . . . . . . . . . . . . . . 256 B.2 Application to asymptotic distributions . . . . . . . . . . 259 B.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 C S-plus functions 261 References 263 Author Index 277 Subject Index 281
List of Tables12.1Polynomials gand Bernoulli numbersB,for asymptotic201expansions.20312.2 Asymptotic expansions for U(2;12,n) distributions.13.1 Asymptotic efficiency of S estimate of scatter at thenormal225distribution. .13.2Asymptotic significancelevel of unadjusted LRTfor α=5%.238
List of Tables 12.1 Polynomials δs and Bernoulli numbers Bs for asymptotic expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . 201 12.2 Asymptotic expansions for U(2; 12, n) distributions. . . . 203 13.1 Asymptotic efficiency of S estimate of scatter at the normal distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 225 13.2 Asymptotic significance level of unadjusted LRT for α = 5%. 238
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List of Figures2.1Bivariate Frank density with standard normal marginals and27a correlation of 0.7.3.11Bivariate Dirichlet density for values of the parameters Pi=41P2 = 1 and p3 = 2.5.1Bivariate normal density for values of the parameters μ=59μ2=0,01=02=1,andp=0.7.5.2Contours of the bivariate normal density for values of theparametersμ1=μ2=0,1=02=1,andp=0.7.Values60of c=1, 2, 3weretaken.615.3A contour of a trivariate normal density.Power function of Hotelling-T2 when p= 3 and n = 40 at a8.1101levelofsignificanceaα=0.05.8.2Power function of the likelihood ratio test for Ho :R = 0113when p =3,and n =20 at alevel of significance α =0.05.11.1 Q-Q plot for a sample of size n= 50 from a trivariate normal.187N3(o,I), distribution.11.2 Q-Q plot for a sample of size n = 50 from a trivariate t on 1188degree of freedom, t3,1(0,I)=Cauchy3(0,I), distribution
List of Figures 2.1 Bivariate Frank density with standard normal marginals and a correlation of 0.7. . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Bivariate Dirichlet density for values of the parameters p1 = p2 = 1 and p3 = 2. . . . . . . . . . . . . . . . . . . . . . . 41 5.1 Bivariate normal density for values of the parameters µ1 = µ2 = 0, σ1 = σ2 = 1, and ρ = 0.7. . . . . . . . . . . . . . . 59 5.2 Contours of the bivariate normal density for values of the parameters µ1 = µ2 = 0, σ1 = σ2 = 1, and ρ = 0.7. Values of c = 1, 2, 3 were taken. . . . . . . . . . . . . . . . . . . 60 5.3 A contour of a trivariate normal density. . . . . . . . . . . 61 8.1 Power function of Hotelling-T2 when p = 3 and n = 40 at a level of significance α = 0.05. . . . . . . . . . . . . . . . . 101 8.2 Power function of the likelihood ratio test for H0 : R = 0 when p = 3, and n = 20 at a level of significance α = 0.05. 113 11.1 Q-Q plot for a sample of size n = 50 from a trivariate normal, N3(0, I), distribution. . . . . . . . . . . . . . . . . . . . . 187 11.2 Q-Q plot for a sample of size n = 50 from a trivariate t on 1 degree of freedom, t3,1(0, I) ≡ Cauchy3(0, I), distribution. 188