Index
- a
- Absolute shrinkage 11, 16, 43, 80, 109, 112, 143, 199, 200, 221, 225, 251, 294, 303
- Actual 308
- Adaptive ridge regression estimator 156–158, 168
- Additional improvements 323
- ADR see Asymptotic distributional risk (ADR)
- Algorithm 158–160, 168, 297, 298
- Alternative approach 33–34
- Alternative expressions 33
- Alternative hypothesis 45, 79
- Alternative PTE 33–34
- Analysis of bioassay 79
- Analysis of ridge regression 11
- Analysis of variance (ANOVA) model 12, 43–79, 109
- Application 12, 60–63, 143–169, 303–324
- Arashi, M. 4, 7, 12, 171, 176, 189, 288, 290
- Artificial intelligence 304
- Assumption 17, 24, 75, 82, 84, 116, 118, 149, 172, 174, 189, 190, 221, 258, 286, 291, 296–300
- Asymptotic distributional bias 202, 226, 259–262
- Asymptotic distributional risk (ADR) 202–213, 226–229, 259–261, 299
- Asymptotic marginal distribution 199–201, 224, 225, 256
- Asymptotic results 166–167
- Autocorrelation coefficient 243–245, 247
- Autoregressive errors 13, 221–250
- Autoregressive model 223
- Autoregressive process 238
- Average neighborhood 182, 184
- b
- Background 315–316
- Bayesian method 15
- Best linear unbiased estimator (BLUE) 45, 152, 154, 161–162
- Bias
- Biased estimator 154
- Big data 13, 303–324
- Binary classification 304, 307, 308, 315–318, 320
- Binary data 8
- Bioassay 79
- Bioinformatics 285
- Biomedical 197
- Biostatistics 197
- Block 44
- Block‐diagonal 44
- Block‐diagonal vector 44
- BLUE see Best linear unbiased estimator (BLUE)
- Bound 51, 85, 115–117, 125, 146, 152, 202, 226, 235, 252, 261, 262, 267, 279
- Boundary 320
- c
- Cancer data 160, 168
- Canonical 2
- Carbon dioxide 237
- Central chi‐square 28, 145
- Central F‐distribution 28, 29, 45
- Characteristic roots 240
- Characteristics 140, 143, 151, 160, 197, 213, 240, 254, 258, 280, 291, 309, 313, 322
- Chemometrics 285
- Chi‐square distribution 28, 45, 114, 145, 191, 200, 224, 226, 255, 259, 282
- Class 43, 72, 79, 96, 129, 268, 320
- Classical shrinkage 43
- Classical shrinkage estimator 43
- Classification 308, 315–318, 320
- Classification problems 304, 307
- Coefficient 1, 4–6, 11, 12, 44, 50, 109, 110, 117, 118, 146, 153–156, 160, 166, 175, 180–182, 193, 200, 221, 225, 251, 261, 285, 290, 291
- Company 197
- Comparison 25–27, 31–33, 43, 44, 109, 206–209, 211–213, 230–236
- Comparison of estimators 52–60, 86–93, 120–127, 165–166, 262–268
- Computational flow graph 308
- Computationally expensive 315, 323
- Concluding remarks 41, 72–74, 96–106, 129–140, 168, 193–194, 213–218, 245–248, 268–281, 300, 323
- Conditional probability 198, 307
- Condition index 237
- Condition numbers 4, 152, 183, 237
- Configuration 8, 49, 50, 202
- Consistent 10, 288
- Contents 72, 96, 129, 268
- Continuous 9, 46, 109, 111, 112, 149, 150, 176, 201, 225, 251, 252, 257
- Converges 159, 290, 300
- Convex 1, 5, 6, 8, 11, 176, 290, 320
- Correlation
- Covariance 221
- Covariance matrix 1, 2, 28, 111, 144, 221–223
- Critical value 8, 25, 28, 29, 47, 61, 83, 146, 162, 176, 194, 200, 224
- Cross‐entropy loss function 309, 310
- Cross validation 179, 192, 194, 316
- Cumulative distribution 30, 48, 83, 114, 202, 226, 252, 308
- Cumulative distribution function 30, 48, 83, 114, 202, 226, 252, 308
- d
- Data 4, 35, 38, 46, 60, 109, 111, 147, 152, 153, 206, 221, 237, 251
- Decision
- Decreasing function 25, 31, 52, 53, 56, 57, 86, 87, 89, 90, 120, 121, 123, 125, 148, 149, 205, 212, 230–233, 235, 262, 263, 265, 266
- Deep neural networks 306, 313–316, 318, 322
- Degrees of freedom (DF) 8, 28, 45, 82, 113, 145, 154–155, 191, 224, 255
- Derivative 19, 148, 150, 309–311, 314
- Design matrix 1–3, 7, 10, 12, 43, 109–111, 146, 173, 179, 187, 251
- Determination 79, 151, 153, 168, 193, 291
- Diagonal matrix 159, 287
- Diagonal projection 43, 50, 52
- Diagram 182, 188, 307, 308, 311
- Dimension 9, 10, 144, 189, 223, 286, 288, 294, 296
- Discrete function 201, 225
- Discuss the contents 72, 96, 129, 268
- Disease 197
- Distributional bias 202, 226, 259–262
- Distributional risk 202
- Distribution function 48, 114, 202, 226, 308
- Divergence 44, 73, 74, 96
- Divergence parameter 44, 73, 74, 96
- Dominance 143
- Dominate 15, 18, 26, 38, 41, 44, 54, 72–74, 87, 96, 129, 165, 166, 209, 212, 213, 241, 248, 268, 293
- Donoho, D.L. 20, 24, 38, 43, 49, 50, 72, 80, 96, 111, 114–117, 200, 256, 257
- e
- Ecological data 285
- Effect 10, 11, 25, 43, 45, 50, 51, 110, 140, 144, 152, 153, 155, 171, 187, 192, 193, 243–245, 280, 306, 311, 313, 319, 320, 322
- Efficiency 31, 35, 53, 63–72, 93–96, 127–129, 213
- Efficiency table 74, 76, 205, 206, 213
- Eigenvalue 2, 145, 146, 148, 152, 179, 182, 187, 287, 296
- Eigenvalue matrix 2, 7, 10, 154
- Eigen vector 2, 3, 285
- Equal 2, 9, 44, 47, 56, 75, 88, 110, 119, 123, 147, 148, 151, 156, 183, 193, 200, 206, 210, 223, 225, 233, 257, 265, 288
- Equation 35, 46, 112, 118, 120, 122, 147, 155–157, 167, 198, 199, 210, 223, 230, 232, 240, 257, 262, 265, 307, 309, 311, 313, 315
- Equivalent 5, 157, 159
- Error 15–17, 24, 41, 44, 82, 110, 143, 222, 245, 252, 300, 320
- Error term 222
- Estimated parameter 222
- Estimated risk 293–295
- Estimation 7, 13, 15–17, 27–28, 44–48, 79–82, 143–145, 161–167, 171–174, 184, 187, 189, 237–240
- Estimation strategy 297–298
- Estimators 5–6, 8–9, 24, 46–60, 81–93, 110–127, 146–151, 155–158, 161–166, 174–178, 199–213, 223–229, 256–268
- Evaluation
- Exercise 24, 191, 197, 313
- Experiment 8, 43, 285, 317
- Experimental results 306, 319–323
- Expressions of the estimators 29, 30, 37, 48, 82–85, 204–213
- g
- Gasoline mileage data 152, 153
- Generalized cross validation 151
- Generalized least squares estimator (GLSE) 175, 222, 227
- Generalized ridge 175
- Generalized ridge regression estimator (GRRE) 155–156, 158, 177, 189
- General linear hypothesis 43
- Global mean temperature 237
- GLSE see Generalized least squares estimator (GLSE)
- Gradients 309–314, 320, 322, 323
- Graph 32, 35, 149, 311
- Graphical representation 54, 87, 264
- GRRE see Generalized ridge regression estimator (GRRE)
- Guarantee 151, 172
- Guaranteed efficiency 35, 36
- h
- Hard threshold estimator (HTE) 46, 48–49, 72, 73, 96, 114–116, 256, 259–252
- Health science 197
- H‐function 202, 226
- High‐dimensional 9–11, 188–193, 285–301
- Higher 44, 305, 306, 312, 315, 319, 322, 323
- Housing prices data 182–189, 194
- HTE see Hard threshold estimator (HTE)
- Hundred samples 305
- Hyper‐parameters 307, 323
- Hypothesis 43, 61, 79–82, 145–146, 239
- i
- Ideal 50–52, 58, 72, 92, 96, 117, 261, 296
- Identity matrix 16, 44, 143
- Ill‐conditioning 4
- Image
- Incorrect signs 201
- Increase 31, 35, 53, 73, 86, 96, 121, 129, 146, 148, 205, 206, 208, 231, 243, 263, 279, 305, 306, 311–314, 320, 322
- Independent 28, 49, 116, 171, 182, 221, 260, 304, 307, 316
- Indicator function 8, 22, 29, 47, 111, 172, 176, 201, 225, 256, 258
- Information matrix 198
- Intercept 27–30, 35–40
- Intercept parameter 252, 253
- Interval 26, 27, 35, 52, 53, 60, 86, 93, 121, 127, 155, 231, 236, 263, 268, 282
- l
- Large coefficients 43, 154
- Large sample 17, 200, 224
- Learning methods 315, 316
- Least absolute shrinkage and selection operator (LASSO) 11, 16, 43, 80, 109, 143, 199, 221, 251, 294, 303
- Least square estimator (LSE) 1, 15, 44, 80, 109, 143, 221, 261, 285
- Level of significance 9, 34–35
- Linear hypothesis 43, 79
- Linear model 15–42, 79–107, 110–114, 172–174
- and rank estimation 252–258
- Linear projecton 50, 117–119, 201, 225, 261–262
- Linear regression model 1, 2, 41, 43, 143, 168, 171–195, 237, 239, 251
- Loan 197
- Local alternative 17, 191, 202, 226, 227, 255, 256
- Location model 12, 15–27, 41
- Logistic regression
- Logit 198, 307, 309, 311, 312
- Log‐likelihood 16, 198
- Loss function 309, 310, 312, 320
- Loss function with penalty 310–311
- Low‐dimensional models 161–167
- Lower 44, 51, 85, 117, 125, 129, 146, 152, 202, 226, 235, 261, 262, 267, 279, 322
- LSE see Least square estimator (LSE)
- m
- Marginal distribution 47, 110–113, 161, 201, 225, 255, 257, 258
- Marginally significant 239
- Mathematical analysis 43
- Matrix 8, 16, 44, 149, 152, 154, 161, 174, 198, 199, 213, 252, 280, 309, 322
- Maximum 25, 31, 35, 36, 206, 207, 211, 231, 233, 266
- Maximum likelihood 198
- Maximum likelihood estimator (MLE) 15, 143, 175, 203
- M‐based xxvii
- Mean 28, 53, 86, 171, 179, 203, 205, 221, 227
- Means squared error (MSE) 9, 16, 48, 82, 147, 258, 288
- Microarray studies 285
- Minimum
- Missing values 237
- Model
- Moderate to strong multicollinearity 237
- Modified
- Monotonically
- Monte Carlo simulation 178, 194, 291–293
- MSE expression 20–24, 30, 31, 34, 36–38, 82–85
- Multicollinearity 1, 3–5, 11, 12, 152, 153, 175, 182, 183, 187, 237–238
- Multiple linear model 6, 10, 110, 143–146, 252, 285, 291, 294
- Multiple regression 12, 13, 109–141, 161, 197
- Multivariate normal 117–119, 261–262
- Multivariate normal distribution 206, 213
- n
- National Oceanographic and Atmospheric Administration 237
- Negative 9, 122, 148, 166, 176, 232, 243, 248, 265, 290, 308, 314, 318
- Negative effect 205
- Neural networks 303–324
- New techniques 309, 310, 315
- Noncentral‐F 28, 30, 31, 146, 176
- Noncentrality parameter 28, 31, 45, 48, 82, 83, 114, 146, 176, 191, 202, 247, 248, 259
- Nonconvergence 310
- Nonlinear 51, 117, 179, 187, 193, 198, 262, 308
- Nonnegative 56, 57, 89, 90, 109, 123–125, 150, 165, 206–209, 211, 212, 233–235, 266
- Nonorthogonality design matrix 251
- Nonspherical 222
- Nonzero effect 296
- Normal 6, 27, 28, 145, 179, 184, 292
- Normal equation 147, 156, 157, 199, 223
- Normalized 44
- NRB 44
- Null distribution 17, 45, 162, 201, 225
- Null hypothesis 16, 28, 29, 41, 45, 61, 79, 111, 145, 176, 185, 190, 241, 248, 258, 288
- Numerical analysis 178–188
- Numerical optimization 309
- o
- Objective function 19, 35, 47, 147, 154–157, 175, 189, 251
- Ocean heat content 237
- Optimization 1, 5, 6, 8, 11, 158–160, 168, 286, 309, 316
- Optimum
- Oracle 43, 50, 117–119, 261–262
- Oracle properties 44, 58, 72, 73, 92, 96, 126, 235, 267
- Orthogonal
- Oscillation Index 237
- Otherwise 29, 31–33, 52, 59, 86, 92, 111, 121, 127, 146, 206, 231, 236, 256, 263, 267, 282, 310
- Outperform 9, 17, 52, 54, 57, 59, 60, 86, 88–90, 92, 93, 96, 121–125, 127, 206–213, 231–236, 241, 263, 265, 267, 268, 279, 282, 290
- Over‐fitting 306, 310, 319, 320, 323
- p
- Parallel 169, 195, 219, 250, 283
- Parameter 2, 9, 15, 44, 79, 152, 213, 296, 310, 322
- Parametric approach xxvii
- Partial derivatives 309, 311
- Partially linear regression 171–195
- Partition 10, 79, 110, 161, 189, 199, 223, 296, 318
- Penalized least squares 297
- Penalty estimator 16, 41, 43, 46–47, 72, 79, 96, 109, 111–114, 129, 168, 193, 197, 199–202, 204, 223–226, 229, 247, 255, 256, 258, 259, 268
- Penalty function 147, 155, 156, 175, 297, 303, 306, 310–312, 314, 319, 320, 323
- Penalty R‐estimators 255–258
- Performance 12, 43, 48, 109, 111, 118, 154, 185, 187, 194, 204, 213, 230, 240, 241, 243, 247, 248, 255, 291, 292, 300
- Performance characteristics 160, 254, 258, 280
- Picture 152, 316–318, 320
- Positive
- Positive‐rule Stein‐type estimator (PRSE) 43, 46, 48, 82, 109, 162, 164, 201, 225
- Positive‐rule Stein‐type generalized ridge estimator (PRSGRE) 176, 180–182, 187, 191, 193
- Possible 16, 35, 118, 153, 315, 317, 319
- Post Selection Shrinkage 293–300
- Prediction 149, 197, 251, 293, 304, 315, 318
- Prediction accuracy 44, 46, 109, 251, 306
- Preliminary test estimator (PTE) 24, 43, 55, 79, 109, 143, 177, 201, 224–225, 251
- Preliminary test M‐estimator 12
- Preliminary test R‐estimator 258
- Preliminary test ridge regression estimator 163–164
- Precision 318
- Principle 111, 257
- Probability density function 22, 49, 179, 252
- Problem of estimation 29
- Proposed estimators 178, 180, 187, 189, 191, 193, 240, 242–244, 246, 248, 249, 291, 292
- Prostate cancer data 160, 168
- PRSE see Positive‐rule Stein‐type estimator (PRSE)
- PRSGRE see Positive‐rule Stein‐type generalized ridge estimator (PRSGRE)
- Pseudorandom numbers 178, 230, 262, 292
- PTE see Preliminary test estimator (PTE)
- P‐value 237
- q
- Qualitative assessment 315
- Quantile‐based xxvii
- r
- Rank‐based LASSO 260–261
- Rank‐based PTE 251, 268, 279
- Rank‐Based Shrinkage Estimation 251–283
- Rank‐based statistics 251
- Rank estimators 251
- Ratio 179
- Real data application 237
- Recall 47, 51, 52, 86, 113, 255, 307, 318
- Region 44
- Regression 1–13–3, 41, 43, 51–52, 109–141, 143–169, 171–195, 197–219, 221–250, 252, 256, 285–301, 303–313, 315–318, 320, 322, 323
- Regression coefficient 4, 5, 111, 143, 153, 237, 239, 256, 294, 296
- Regularity conditions 49, 50, 172–174, 191, 202, 226, 227, 256, 260, 261, 286, 288, 296, 297
- Regularization 251, 311, 314
- Regularization parameter 10
- Relative
- Relative weighted L2‐risk efficiency (RWRE) 52–56, 59, 61, 64–71, 73, 75, 86–88, 93, 97–106, 120–123, 262–265, 269–281
- Residual sum of squares 1, 5, 148, 193, 291
- Response 1, 43, 143, 171, 172, 192, 197, 221, 285
- Response vector 110
- R‐estimate 251, 281
- R‐estimator 251, 252, 254–262, 264, 268, 271–274
- Restricted estimator 8, 29, 31, 38, 47, 54, 72, 96, 113, 119, 129, 168, 173, 185, 189, 203, 213, 241, 245, 268, 297, 300
- Restricted generalized least squares estimator (GLSE) 227
- Restricted least squares estimator (RLSE) 29, 41, 45, 52, 54–56, 58–59, 81, 86–89, 92, 109, 120, 122–123, 125–126
- Restricted maximum likelihood estimator (RMLE) 199, 206, 209–210
- Restricted M‐estimator 12
- Restricted R‐estimator 258, 262, 268
- Restricted ridge regression estimator 161
- Riboflavin data 192–194, 291, 292, 300
- Ridge
- Ridge regression estimator (RRE) 5–6, 16, 46, 51–52, 85, 109, 119, 146–151, 155–158, 161–162, 168, 177, 189, 199, 221, 251, 285, 300
- Ridge‐type shrinkage estimation 35
- Risk
- Risk difference 56, 57, 89, 90, 123–125, 165, 206, 210–212, 233–235
- Risk efficiency 63, 93, 204–213, 230–236
- Risk function 57, 90, 111, 125, 203, 213, 227–229, 240, 241
- RLSE see Restricted least squares estimator (RLSE)
- RMLE see Restricted maximum likelihood estimator (RMLE)
- Robust ridge 12
- RWRE see Relative weighted L2‐risk efficiency (RWRE)
- s
- Saleh, A.K.M.E. 8, 9, 12, 24, 29, 43, 80, 111, 112, 175–177, 251, 255, 257
- SE see Stein‐type estimator (SE)
- Sea level 239
- Sea level rise 237–245
- Seemingly unrelated 12, 79–107
- Selection operator 11, 16, 43, 80, 109, 112, 143, 199, 200, 221, 225, 251, 294, 303
- Sequence of local alternatives 17, 202, 226, 227, 255
- Serial correlation 221
- SGRRE 187, 193
- Shallow neural network 312, 315, 316, 318, 323
- Shrinkage estimation 17, 251–283, 297
- Shrinkage estimator (SE) 9, 17, 43, 46, 113–114, 119–120, 168, 171, 176–178, 190, 200–202, 224–226, 254, 288, 300
- Sign function 21, 239
- Significant 44, 111, 193, 197, 221, 237, 239, 256, 303, 320
- Simple linear model 12, 15–42, 79–107
- Simulation 43, 44, 111, 182
- Singular value decomposition 7, 154
- Slope 15, 27, 35–40, 79, 192
- Slope parameter 27–30, 41
- Soft threshold estimator 20, 46, 118
- Sparse 10, 47, 50–52, 58, 72, 79, 92, 96, 112, 115–117, 125, 193, 235, 251, 257, 260–262, 267
- Sparse model 44, 110, 298
- Sparsity 10, 110, 113, 189, 199, 223, 255, 292, 298
- Sparsity condition 113, 161, 172, 200, 224, 254, 258
- Spectral decomposition 2
- Standardized 178, 237, 292
- Standardized distance 31
- Standard logistic 305, 306, 315
- Statistical learning methods 315
- Statistical methods 285, 303, 304
- Statistical technique 315
- Stein estimator 43, 46, 47, 119
- Stein‐type 55, 58, 59, 61, 88, 91, 93, 125, 126, 143, 177, 178, 280
- Stein‐type estimator (SE) 1, 12, 25, 33, 34, 43, 44, 47–48, 72, 79, 81–82, 109, 111, 129, 168, 202–204, 225–229, 245, 251, 257, 268, 288, 300
- Strong correlation 298
- Subset selection 43, 44, 109, 154, 155, 201, 251, 259–260
- Sunspots 237
- Supervised learning methods 316
- Suspect 18, 19, 44, 79, 171, 200, 223, 224
- Suspect sparsity 161, 199, 223, 258
- t
- Table 26, 43, 62, 74, 76, 205, 206, 213
- Temperature variable 239
- Test
- Testing for multicollinearity 237–238
- Theoretical statistics 4, 12, 72, 96, 109, 129, 268
- Theory of ridge 11
- Total sparse 44
- Total sparse mode 44
- Trace of a matrix 154
- Traditional estimator 31
- Training data 315, 316, 319, 320
- Treatment 43–45, 50, 51
- t‐test 4
- Tuning Parameters 6, 147, 148, 151, 153, 157, 168, 286, 307, 309, 311, 320
- Two‐layer logistic regression 311–313
- v
- Variables 3, 9–11, 120, 152–155, 178, 182, 197, 230, 237–239, 262, 286, 292, 293, 300, 304–308, 310, 311, 314, 320
- Variance 2, 3, 5, 16, 44, 79, 110, 146, 148, 149, 166, 171, 179, 222, 286
- Variance‐covariance 2, 222
- Variance inflation factor (VIF) 4, 153, 237, 238
- Vector of responses 1, 143, 285
- VIF see Variance inflation factor (VIF)
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