k

Index

Affine equivariance 203 robust M-estimates of the Angle correlation coefficient 166 Mahalanobis 190 robust principal component Association 1, 5, 10, 29 correlation coefficient 158 Granger’s dependence measure of robust principal component 30 correlation informational measures of 29 coefficient–regularity Joe’s dependence measure of 29 conditions 159 measures of 5, 10 sample correlation coefficient 148 Silvey’s dependence measure of 29 k Asymptotic Bias k FQ estimate 122 highly efficient robust estimate of 𝛼 FQn estimate 122 the correlation coefficient 𝛼 MQn estimate 122 164 rK Kendall robust principal component coefficient 152 correlation coefficient 160 rMAD MAD correlation coefficient sample correlation coefficient 148 161 Bivariate normal distribution 3, 4, 13, rQ quadrant (sign) estimate 152 22 rS Spearman rank correlation Boxplot 231 coefficient 153 bagplot 231 highly efficient robust estimate of hist-plot 233 the correlation coefficient notched boxplot 232 164 Tukey’s boxplot 231, 232 maximumCOPYRIGHTED likelihood estimate of the univariateMATERIAL boxplots 232 correlation coefficient 149 vaseplot 233 radical stable estimate of the Breakdown point 52 correlation coefficient 174 FQ estimate 124

Robust Correlation: Theory and Applications, First Edition. Georgy L. Shevlyakov and Hannu Oja. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd. Companion Website: www.wiley.com/go/Shevlyakov/Robust

k k

312 INDEX

Breakdown point (continued) rMMAD M-estimate of 168 𝛼 FQn estimate 124 rMMAD M-estimate of the correlation 𝛼 MQn estimate 124 coefficient 170 Fourier transform rQn correlation coefficient 146 periodogram 264 rQ quadrant (sign) correlation coefficient 170 Canonical rQ quadrant (sign) estimate of 152 correlation 195 rS Spearman correlation coefficient Canonical correlations and variables 170 asymptotic distribution 196 rTRIM trimmed correlation Cauchy coefficient 147 multivariate model 185 rmed median correlation coefficient Change-of-variance function 75 146 Coefficient between random events 23 correlation 181 coefficient of determination 16 Coefficient of association 4 comedian correlation coefficient 143 Coefficient of contingency 4 correlation ratio 16 Coefficient of determination 4 direct robust counterparts of Commutation Pearson’s correlation matrix 179 coefficient 142 Component analysis distance correlation coefficient 12, principal 191 k 29 k Copula Gebelein’s correlation coefficient 29 multivariate model 186 Gebelein’s correlationcoefficient 11 Correlation1,2,5,10 highly efficient robust estimates of analysis 2 163 history 1 measures of 5, 10 identity 20 requirements 10 informational correlation coefficient sign and rank 221, 224, 227, 229 30 Correlation analysis 4 Kendall rank correlation coefficient canonical 195 4, 24, 28, 143, 152 Correlation coefficient 1, 4, 10 maximal correlation coefficient 11 maximum likelihood estimate of rFQ correlation coefficient 146, 170, 260 149 maximum likelihood estimate of the rK Kendall correlation coefficient 170 correlation coefficient in rL2 correlation coefficient 147 the ICD model 165 rMAD MAD correlation coefficient minimax variance M-estimate of 146, 170, 260 167 rMFQ M-estimate of 168 nonparametric estimates of 151 rMFQ M-estimate of the correlation Pearson’s correlation coefficient 3, coefficient 170 10–12, 140, 148

k k

INDEX 313

Pearson’s correlation coefficient via Data spectrum 5 13 measures of 5 quadrant (sign) correlation Data 5 coefficient 24, 26, 143 types of 5 radical stable estimate of 173 Definition robust M-estimates of 165 breakdown point 210 robust minimax bias estimate of 161 influence function 205 robust minimax variance estimate of Detection of 231 161 boxplots 231 robust principal component Determinant variances correlation matrix 205 coefficient 146 Distance rm1 correlation Mahalanobis 190 coefficient 144 Distribution robust regression rm2 correlation bivariate Cauchy (t-distribution) 168 coefficient 144 bivariate Cauchy ICD 156, 168 robust regression LAV correlation bivariate normal 168 coefficient 143 bivariate normal ICD 156 robust regression LMS correlation Cauchy 290 coefficient 144 Cauchy contaminated standard sample correlation coefficient 3, 4 normal 290 k Sarmanov’s correlation coefficient Cauchy noise 279 k 11, 29 contaminated bivariate normal 168 Spearman rank correlation contaminated Gaussian noise 279 coefficient 11, 24, 27, Gaussian noise 279 143, 153 generalized Gaussian noise 279 two-stage procedures robust independent component correlation coefficient 147 distributions 155 via the cosine of the angle between Laplace 290 the variable vectors 21 Laplacian noise 279 via the ratio of two 22 standard normal 290 via the variances of the principal Tukey gross-error and ICD models components 18 156 Covariance Distribution (or distribution density) sign and rank 221, 224, 225, 228, 37, 58, 87 229 Cauchy 78 Covariance matrix 180 Cauchy contaminated normal 78 Cauchy distribution 131 Dümbgen exponential- power 79 estimate 210, 226 Laplace 78 Data analysis 6, 108 lattice distribution 69 aims of 6 least informative (or least favorable) estimation of scale 108 37

k k

314 INDEX

Distribution (or distribution density) Eigenvectors and eigenvalues of (continued) sample covariance matrix least informative approximately asymptotic distribution 195 finite (cosine– Elliptical exponential) 60 multivariate model 185 least informative finite (cosine Entropy 29 form) 59 mutual information 30 least informative lattice distribution relative 29 72 Shannon’s entropy 29 least informative nondegenerate Euclidean (Laplace) 58 norm 222 least informative with a bounded Euler equation 39 variance (normal) 58 Exchangeability 183 least informative with a bounded Exploratory data analysis 231 variance and distribution boxplots 231 density at the center of Extreme values 5 symmetry 63 measures of 5 least informative with bounded subranges 65 Fisher information 70 standard normal 78, 87 lattice distribution analog 70 Tukey gross error model 131 maximum likelihood estimate of the k Weber-Hermite distribution correlation coefficient 149 k densities 62 Fisher transformation 148 Distribution classes Forms of data representation 5 approximately finite distributions 58 finite distributions 57 Gaussian distribution (or normal) 87 nondegenerate distributions 57 Berry-Esseen inequality 96 with a bounded variance 57 CLT 93 convolution of Gaussians 92 derivation of Gauss 87 derivation of Herschel and Maxwell FQ estimate 124 89 𝛼 FQn estimate 124 derivation of Landon 90 𝛼 MQn estimate 124 Fourier transform of a Gaussian 93 rK Kendall rank correlation maximizing entropy 97 coefficient 152 minimizing Fisher information 99 rQ quadrant (sign) estimate 152 properties 92 rS Spearman rank correlation Generalized variance 205 coefficient 153 Geometric lattice distribution 71 maximum likelihood estimate of the Fisher information of 71 correlation coefficient 149 radical stable estimate of the Hettmansperger-Randles correlation coefficient 174 estimate 210, 226

k k

INDEX 315

Huber Location 5, 10, 33, 36 M-functional and estimate 208 L-estimates of 44 M-estimates of 36 ICD models 308 R-estimates of 45 Independence asymptotic equivalency of M-, L- test for 197 and R-estimates of 46 Independence between subvectors asymptotic normality 38 test for 197 asymptotic variance of M-estimates Independent component 36 multivariate model 186 cosine extremals of the variational Independent component analysis 186, problem 41 214 distribution density class 37 Independent component distributions efficiency of M-estimates 37 155 exponential extremals of the properties 155 variational problem 41 Influence function 49, 122 Fisher information for 36, 37 FQ estimate 122 Hampel’s redescending 25A FQ𝛼 estimate 122 n three-part score estimate MQ𝛼 estimate 122 n of 80 Q estimate 122 Huber’s linear bounded score r Kendall rank correlation K estimate of 80 coefficient 152 k Huber’s optimal score function 42 k r quadrant (sign) estimate 152 Q maximin stable redescending r Spearman rank correlation S M coefficient 153 -estimate of 83 one-step FQ-estimate 130 maximum likelihood estimate of 80 𝛼 measures of 5, 10, 33 one-step FQn -estimate 130 sample correlation coefficient 148 minimax bias M-estimates of 43 sample 49 minimax score function of sample median 49 M-estimate of 37 Interdependence 1, 5, 10 minimax variance M-estimates of 36 measures of 5, 10 minimum sensitivity stable estimate Invariant coordinate selection 214 of 80 multivariate 203 Kendall rank correlation coefficient optimality condition for the least 221, 224 informative distribution Kronecker product 179 40 radical stable estimate of 80 multivariate 214 regularity conditions 38 sample mean 80 L1 criterion function 219 sample median 80, 260 Laplace score function of M-estimates of 36 multivariate model 186 score functions class 37

k k

316 INDEX

Location (continued) probability-free approach 7 Smith’s stable redescending Monte Carlo 131, 264, M-estimate of 84 278, 289 Tukey’s biweight stable FQ-estimate 131 redescending M-estimate Q-estimate 131 of 84 accuracy of 131 variational problem of minimizing additive contamination 264 Fisher information for 39 disorder contamination model 264 Location-scatter median of absolute deviations 131 multivariate model 183 one-step M-estimate FQ 131 robust estimates of scale M-estimates performance 131 influence function 209 robust estimates of the correlation M-estimates of multivariate location, coefficient 168 scatter and correlation robust fusion detection 303 asymptotic distribution 209 131 Manhattan Multivariate location norm 220 M-functional and estimate 208 Mardia’s measures of and Multivariate location and scatter kurtosis estimates multivariate 191 asymptotic distribution 207 k Marginal Multivariate scatter k sign and rank 220 M-functional and estimate 208 Matrix Multivariate statistical analysis 4 norm 179 robust correlation 212 Nonparametric methods Maximum likelihood estimate 3 periodogram 255 correlation coefficient 3 power spectrum definition 256 Maximum likelihood method 4 5, 24 Mean deviation 219 measures of correlation 24 mean difference 219 nonparametric facts of statistics 25 Median 26 Norm multivariate 219, 225, 228 matrix 179 sample median 26 Normal Methods of estimation 7 multivariate model 184 Bayesian methods 7 classical parametric methods 7 Oja exploratory data analysis methods 7 sign and rank 226 fuzzy methods 7 geometrical methods 7 Parametric interval probability methods 7 multivariate model 183 logical-algebraic methods 7 Partial nonparametric methods 7 correlation 199

k k

INDEX 317

Perception ability 6 robust cross-product DFT analogs psychological bounds on 6 261 Pillai’s trace 199 robust estimation 255, 259, 309 Points of growth 308 robust estimation of the applications 309 Yule-Walker open problems in multivariate autoregressive parameters statistics 308 263 robust estimation of a correlation robust filtering 263 matrix in ICD models 308 robust periodograms 262 robust estimation of a power robust versions of the spectrum 309 Blackman–Tukey robust estimation of eigenvalues and formula 262 eigenvectors in ICD robust versions of the Fourier models 308 transform 309 stable estimation of a correlation matrix 308 Rao–Cramér inequality 70 stable estimation of multivariate lattice analog 70 location 308 Redescending M-estimates 53 stable estimation of multivariate biweight estimate 54 scale 308 Hampel’s 25A estimate 54 Postulates 10, 12 Huber’s skipped mean 54 k correlation 10 Smith’s estimate 55 k correlation Renyi 12 Regression 1, 2, 16 Renyi 10, 30 explained sum of squares 17 Schweizer and Wolff 10 linear regression 16 Power exponential 16 multivariate model 186 total sum of squares 17 Power spectrum 255, 309 Robust detection 272 Lp -normanalogs of the DFT 309 asymptotic fusion rule 298 autoregressive power spectrum 257 asymptotic fusion rule performance Blackman–Tukey formula 255, 259 299 median discrete Fourier transmorm asymptotic fusion rule–numerical 261, 309 results 303 median periodogram 262 asymptotic robust minimax parametric estimation 259 detection of a known periodogram 258 signal 276 robust Lp -norm analogs of the DFT asymptotically optimal robust 261 detection 274 robust analogs of the discrete asymptotically optimal robust Fourier transform (DFT) detection–regularity 261, 309 conditions 275 robust 262 classical approach to detection 272 robust autocovariances 263 detection error sensitivity 288

k k

318 INDEX

Robust detection (continued) of what 33 detection error stability 288 Prokhorov metric supermodel 35 detection stability 291 qualitative robustness 7 Hampel’s redescending three-part quantitative robustness 7 score 289 robust estimation of a power Huber’s linear bounded score 289 spectrum 309 maximum likelihood score 289 robust power spectrum estimation minimax 𝜌-distance rule 279 255 minimax approach to hypothesis stable detection 285 testing 273 stable estimation 73, 173, 285 minimum error sensitivity stable supermodel 33 redescending score 289 Tukey’s supermodel (gross error non-Gaussian noise 309 model) 34 Pitman’s asymptotic efficiency 291 Tukey’s supermodel based on the probability of missing 291 quantile function 34 radical stable redescending score variance sensitivity 74 289 robust fusion detection 296 Sample correlation matrix 182 robust minimax detection based on asymptotic distribution 188 a distance rule 275 Sample covariance matrix 182 sign score 289 asymptotic distribution 187 k stable detection 285 Sample mean vector k weak multivariate signal 309 asymptotic distribution 187 5, 33 Scale 5, 10, 107 Robustness 6, 33 FQ estimate of 122 𝛼 B-robustness 52 FQn estimate of 122 𝛼 V-robustness 52 MQn estimate of 122 against what 33 Q estimate of 121 Andrews supermodel 34 L-estimates of 115 Bickel and Lehmann supermodel 34 M-estimate of 111 estimates of correlation coefficients pth power deviation 109 140 pth-power deviation 112 estimation stability 76 asymptotic normality 114 global measures of 52, 74 Fisher consistency 114 Hampel’s approach 7, 47 Fisher information for 117 Hodges and Lehmann supermodel highly efficient robust estimates of 34 119 Huber’s minimax approach 7, 35, 57 Hodges-Lehmann estimate of 121 in what sense 33 Huber minimax variance estimates Lévy metric supermodel 35 of 116 local measures of 51 least informative distribution 117 of the sample mean 33 least median squares (LMS) of the sample median 33 deviation 109

k k

INDEX 319

maximum likelihood estimate of multivariate 214 113 Spatial mean absolute deviation 108, 112, quantile function 224 116 sign and rank 222 measures defined by functionals 110 Spearman rank correlation coefficient measures of 5, 10, 107 221, 225, 229 median absolute deviation 112 Stability 76, 174 median of absolute deviations 120, maximum likelihood estimate of 260 location 76 minimax variance L-estimates of minimum sensitivity stable estimate 118 of location 76 minimax variance M-estimates of radical stable estimate of location 76 118 radical stable estimate of the one-step FQ-estimate of 124 correlation coefficient 174 𝛼 one-step FQn -estimate of 124 Stable estimation one-step M-estimate of 124 of correlation 308 regularity conditions 114 of location 308 sample 109 of scale 308 standard deviation 108, 112, 116 Standardization 213 trimmed mean absolute deviation 6 115 stability of 6 k trimmed standard deviation 115 Symmetry k variational problem of minimizing multivariate 183 Fisher information 117 Scatter 5, 107 t-distribution measures of 5, 107 multivariate model 185 multivariate 203 Tolerance region 190 Score function Trace FQ estimate 122 matrix 179, 205 𝛼 Tyler FQn estimate 122 𝛼 estimate 210, 225 MQn estimate 122 one-step FQ-estimate 126 Univariate one-step FQ𝛼-estimate 126 n sign and rank 218 Semiparametric multivariate model 184 Values Sensitivity curve 47 principal 192 sample mean 47 Variables sample median 47 canonical 195 Shape principal 192 multivariate 203 vec operation Sign and rank matrix 178 multivariate 220, 222, 226 Skewness Wilk’s test 198

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k k

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