Theory of Preliminary Test and Stein-Type Estimation with Applications

Theory of Preliminary Test and Stein-Type Estimation with Applications......Page 4
Contents......Page 10
List of Figures......Page 20
List of Tables......Page 22
Preface......Page 24
1 Introduction......Page 28
1.2 Statistical Decision Principle......Page 33
1.3 Quadratic Loss Function......Page 35
1.4.1 Mean and Simple Linear Models......Page 36
1.4.2 One-Sample Multivariate Model......Page 39
1.4.3 ANOVA Models......Page 40
1.4.4 Parallelism Models......Page 41
1.4.5 Multiple Regression Model and General Linear Hypothesis......Page 42
1.4.6 Simple Multivariate Linear Model......Page 46
1.4.7 Discrete Data Models......Page 47
1.7 Problems......Page 50
2.1 Normal Distribution......Page 56
2.2 Chi-square Distribution and Properties......Page 57
2.3 Some Results from Multivariate Normal Theory......Page 60
2.4 Beta Distribution and Applications......Page 61
2.5.1 Binomial Distribution......Page 63
2.5.2 Multinomial Distribution......Page 64
2.6 Matrix Results......Page 65
2.7 Large Sample Theory......Page 67
2.7.1 Four Types of Convergence......Page 68
2.7.2 Law of Large Numbers......Page 69
2.7.3 Central Limit Theorems......Page 70
2.8.1 Order-Statistics, Ranks, and Sign Statistics......Page 72
2.8.2 Linear rank-statistics (LRS)......Page 73
2.8.3 Rank Estimators of the Parameters of Various Models......Page 77
2.9 Problems......Page 80
3 Preliminary Test Estimation......Page 82
3.1.2 Estimation of the Intercept and Slope Parameter......Page 83
3.2 PTE of the Intercept Parameter......Page 84
3.2.2 Bias and MSE Expressions......Page 85
3.2.3 Comparison of bias and mse functions......Page 88
3.2.2 Graph of MRE(tn; tn) and MRE(tPTn; tn)......Page 90
3.3.2 Estimation and Test of the Difference between Two Means......Page 94
3.3.3 Bias and mse Expression of the Three Estimators of a Mean......Page 95
3.3.1 Graph of MRE (m1; m1) and MRE(mPT1; m1)......Page 100
3.4.3 Bias, mse, and Analysis of Efficiency......Page 101
3.4.1 Minimum and Maximum Efficiency of PTE......Page 103
3.5.2 One-Sample Problem......Page 104
3.5.3 Comparison of PTE, tPTn and SE tSn......Page 107
3.5.1 Graph of the relative efficiency of SE and PTE for different values of a......Page 109
3.5.5 The Two-Sample Problem and Shrinkage Estimation......Page 113
3.5.3 Minimum and Maximum Relative Efficiency of SE and PTE for a = 0.05(0.10)0.45 and for Selected Samples......Page 115
3.6.2 Conditions for Asymptotic Normality of the Unrestricted Estimators of Intercept and Slope Parameters......Page 116
3.6.3 Asymptotic Distributional Bias and Mean Square Error Expressions, and Efficiency Analysis......Page 119
3.7 Two-Sample Problem and Estimation of Mean......Page 126
3.8 One-Sample Problem and Estimation of the Mean......Page 128
3.9 Stein Estimation of Variance: One-Sample Problem......Page 130
3.10.1 Model and Assumptions......Page 136
3.10.2 Test of Hypothesis......Page 137
3.10.3 Estimation of Intercept and Slope Parameters......Page 138
3.10.4 Asymptotic Distribution of Various Estimators and Their ADB and ADMSE Expressions......Page 139
3.12 Problems......Page 145
4 Stein-Type Estimation......Page 152
4.1 Statistical Model, Estimation, and Tests......Page 153
4.2 Preliminary Test Estimation......Page 156
4.2.2 Maximum and Minimum Guaranteed risk Based Efficiencies......Page 163
4.3.2 James-Stein Estimator (JSE)......Page 166
4.3.3 Positive-Rule Stein Estimator (PRSE)......Page 170
4.3.1 Risk Gain of PRSE over JSE......Page 175
4.4.1 Risk Difference Representation Approach......Page 178
4.4.2 Empirical Bayes Estimation (EBE) Approach......Page 181
4.4.3 Quasi-empirical Bayes or Preliminary Test Estimation Approach......Page 184
4.4.4 How Close is the JS Estimator to the Bayes Estimator?......Page 186
4.5.2 Preliminary Test and Stein-Type Estimators......Page 188
4.5.3 Empirical Bayes Estimation When the Variance Is Unknown......Page 189
4.5.4 Bias, MSE Matrices, and Risk Expressions......Page 190
4.5.5 Risk Analysis of the Estimators......Page 193
4.5.6 An Alternative Improved Estimation of t......Page 198
4.6.1 Model, Estimation, and Test......Page 201
4.6.3 Asymptotic Distributional Bias Vector, Quadratic Bias, MSE Matrix, and Risk Expressions of the Estimators......Page 202
4.7.2 Improving tSn via PTE......Page 207
4.7.3 Iterative PTE to Obtain an Admissible Estimator......Page 209
4.7.4 Extension to the Case Where the Variance Is Unknown......Page 210
4.8.1 Introduction......Page 212
4.8.2 Properties of the Recentered Confidence Set Based on PRSE......Page 214
4.8.2 Some Upper Bounds of c for g = . 10 and . 05......Page 219
4.8.4 Coverage Probabilities for the Set CPT(tPTn(a)) with g = 0.10 and a = 0.10......Page 222
4.9 Nonparametric Methods: R-Estimation......Page 224
4.9.2 Test of Hypothesis......Page 225
4.9.3 Estimation of the Location Parameter......Page 226
4.9.4 ADB, ADQB, ADMSE, and ADQR of the Estimators of Location Parameters......Page 227
4.9.5 Asymptotic Properties of Confidence Sets......Page 231
4.10.1 Properties of Estimators......Page 232
4.11 Problems......Page 233
5 ANOVA Model......Page 240
5.1.2 Estimation of the Parameters of the One-way ANOVA Model......Page 241
5.1.3 Test of Equality of the Treatment Means......Page 242
5.2.1 Preliminary Test Approach (or Quasi-empirical Bayes Approach)......Page 245
5.2.2 Bayes and Empirical Bayes Estimators of Treatment Means......Page 246
5.3.1 Bias Expressions......Page 248
5.3.2 MSE Matrix and Risk Expressions......Page 251
5.4.1 Comparison of tn and tn......Page 256
5.4.2 Comparison of tPTn and tn (tn)......Page 257
5.4.3 Comparison of tSn, tS+n, and tn......Page 259
5.5.1 Comparison of tn and tn......Page 261
5.5.2 Comparison of tPTn Relative to tn and tn......Page 262
5.5.3 Comparison of tn and tSn (tSn and tS+n)......Page 264
5.6 Improving the PTE......Page 267
5.7 ANOVA Model: Nonnormal Errors......Page 269
5.7.1 Estimation and Test of Hypothesis......Page 270
5.8 ADB, ADQB, ADMSE, and ADQR of the Estimators......Page 271
5.8.1 Asymptotic Distribution of the Estimators under Fixed Alternatives......Page 272
5.8.2 Asymptotic Distribution of the Estimators under Local Alternatives......Page 273
5.8.3 ADB, ADQB, MSE-Matrices, and ADQR of tPTn tSn and tS+n......Page 275
5.9 Confidence Set Estimation......Page 277
5.9.1 Confidence Sets and Coverage Probabilities......Page 278
5.9.2 Analysis of the Confidence Sets......Page 280
5.10.1 Asymptotic Representations of Normalized Estimators under Fixed Alternatives......Page 285
5.10.2 Asymptotic Coverage Probability of the Confidence Sets under Local Alternatives......Page 286
5.11.1 Model, Assumptions, and Linear Rank Statistics (LRS)......Page 287
5.11.3 Asymptotic Distributional Properties of R-Estimators......Page 290
5.11.4 ADB, ADQB, ADMSE, and ADQR......Page 292
5.13 Problems......Page 295
6 Parallelism Model......Page 298
6.1.2 Estimation of the Intercept and Slope Parameters......Page 299
6.1.3 Test of Parallelism......Page 301
6.2 Preliminary Test and Stein-Type Estimators......Page 302
6.2.1 The Estimators of Intercepts and Slopes......Page 303
6.2.2 Bayes and Empirical Bayes Estimators of Intercepts and Slopes......Page 305
6.3.2 Restricted Estimators of b and t......Page 307
6.3.4 James-Stein-type Estimators of b and t......Page 308
6.3.5 Positive-Rule Stein Estimators of b and t......Page 309
6.4.1 Bias Comparison of the Estimators of the Intercept Parameter......Page 310
6.4.2 MSE-matrix Comparisons......Page 311
6.4.3 Weighted Risk Comparisons of the Estimators......Page 316
6.5 Estimation of the Regression Parameters: Nonnormal Errors......Page 319
6.5.1 Unrestricted, Restricted, Preliminary Test, James-Stein and Positive-Rule Stein Estimators and Test of Hypothesis......Page 320
6.5.2 Conditions for Asymptotic Properties of the Estimators and Their Distributions......Page 321
6.5.3 Asymptotic Distributions of the Estimators......Page 322
6.5.4 Expressions for ADB, ADQB, ADMSE, and ADQR of the Estimators......Page 325
6.6.2 Comparison of tPTn and tn (tn)......Page 329
6.6.3 Comparison of tSn and tn (tn)......Page 331
6.6.5 Comparison of tS+n and tSn, tn, tPTn......Page 332
6.7 Asymptotic Distributional MSE-matrix Properties......Page 334
6.8.1 Confidence Sets for the Slope Parameters......Page 338
6.8.2 Analysis of Coverage Probabilities......Page 341
6.8.3 Confidence Sets for the Intercept Parameters when s2 is Known......Page 345
6.9 Confidence Set Estimation: Nonnormal Case......Page 346
6.10.1 Model, Assumptions, and Linear Rank Statistics......Page 348
6.10.3 Estimation of the Intercepts ta and the Slope ba......Page 350
6.10.4 Asymptotic Distribution of the R-Estimators of the Slope Vector......Page 352
6.10.5 Asymptotic Distributional Properties of the R-Estimators of Intercepts......Page 357
6.10.6 Confidence Sets for Intercept and Slope Parameters......Page 363
6.12 Problems......Page 364
7 Multiple Regression Model......Page 366
7.1.1 Estimation of Regression Parameters of the Model......Page 367
7.1.2 Test of the Null Hypothesis, Hb = h......Page 368
7.2.1 Preliminary Test (or Quasi-empirical Bayes) Approach......Page 370
7.2.2 Bayes and Empirical Bayes Estimators of the Regression Parameters......Page 371
7.3.1 Bias Expressions......Page 375
7.3.2 MSE Matrices and Weighted Risks of the Estimators......Page 377
7.4 Risk Analysis of the Estimators......Page 382
7.5 MSE-Matrix Analysis of the Estimators......Page 389
7.6 Improving the PTE......Page 396
7.7.2 Estimation of Regression Parameters and Test of the Hypothesis......Page 397
7.8 Asymptotic Distribution of the Estimators......Page 398
7.8.1 Asymptotic Distribution of the Estimators under Fixed Alternatives......Page 399
7.8.2 Asymptotic Distribution of the Estimators under Local Alternatives, and ADB, ADQB, ADMSE, and ADQR......Page 401
7.8.3 ADQR Analysis......Page 407
7.9.1 Preliminaries......Page 410
7.9.2 Confidence Sets and the Coverage Probabilities......Page 412
7.9.3 Analysis of the Coverage Probabilities......Page 414
7.10.1 Confidence Sets......Page 418
7.10.2 Asymptotic Properties of Confidence Sets......Page 419
7.11.1 Linear Rank Statistics, R-Estimators and Confidence Sets......Page 421
7.11.2 Asymptotic Distributional Properties of the R-estimators......Page 423
7.11.3 Asymptotic Properties of the Recentered Confidence Sets Based on R-Estimators......Page 426
7.13 Problems......Page 427
8 Regression Model: Stochastic Subspace......Page 430
8.1.1 The Model Formulation......Page 431
8.1.2 Mixed Model Estimation......Page 432
8.1.3 Test of Hypothesis......Page 433
8.1.4 Preliminary Test and Stein-type Mixed Estimators......Page 434
8.2.1 Bias and Quadratic Bias Expressions......Page 435
8.2.2 MSE Matrix and Risk Expressions......Page 436
8.2.3 MSE Matrix Comparisons of the Estimators......Page 438
8.2.4 Risk Comparisons of the Estimations......Page 442
8.3 Estimation with Prior Information......Page 445
8.3.1 Estimation of b1 and Test of H0b0 = H1b1......Page 446
8.3.2 The Mixed Estimators......Page 447
8.3.4 MSE Matrix and Risk Expressions......Page 448
8.4.1 Introduction......Page 449
8.4.2 Estimation of the Parameters and Test of Hypothesis......Page 450
8.5 Asymptotic Distribution of the Estimators......Page 451
8.5.1 Asymptotic Distribution of the Estimators under Fixed Alternatives......Page 452
8.5.2 Asymptotic Distribution of the Estimators under Local Alternatives......Page 454
8.6 Confidence Set Estimation: Stochastic Hypothesis......Page 456
8.7 R-Estimation: Stochastic Hypothesis......Page 457
8.9 Problems......Page 463
9 Ridge Regression......Page 466
9.1.1 Ridge Regression with Normal Errors......Page 468
9.1.2 Nonparametric Ridge Regression Estimators......Page 469
9.2 Ridge Regression as Bayesian Regression Estimators......Page 470
9.3.1 Bias Vector of bPTn (k)......Page 471
9.4 Covariance, MSE Matrix, and Risk Functions......Page 473
9.5 Performance of Estimators......Page 477
9.6 Estimation of the Ridge Parameter......Page 488
9.8 Problems......Page 491
10 Regression Models with Autocorrelated Errors......Page 496
10.1.1 Estimation of the Intercept and Slope Parameters when r is Known......Page 497
10.1.2 Preliminary Test and S-Estimation of b and t......Page 499
10.1.3 Estimation of the Intercept and Slope Parameters When Autocorrelation Is Unknown......Page 501
10.2 Multiple Regression Model with Autocorrelation......Page 505
10.2.2 Preliminary Test, James-Stein and Positive-Rule Stein- Type Estimators of b......Page 506
10.3 Bias, MSE Matrices, and the Risk of Estimators When r Is Known......Page 507
10.4 ADB, ADMSE, and ADQR of the Estimators (r Unknown)......Page 510
10.5.1 Preliminary Test and Stein-Type Estimators (Chen and Saleh, 1993)......Page 512
10.5.3 Empirical Results and Conclusions......Page 514
10.4.12 Empirical Risk Values for Shrinkage PTE Based on D–W and G1 Statistic, a = 0.05......Page 520
10.6.1 Estimation and Test of Hypothesis......Page 521
10.6.2 Asymptotic Theory of the Estimators and the Test-Statistics......Page 522
10.6.3 ADB, ADMSE Matrices, and ADQR of the Estimators......Page 524
10.7 R-Estimation of the Parameters of the AR[p]-Models......Page 525
10.7.1 R-Estimation of the Parameters of the AR[p] Model......Page 526
10.7.2 Tests of Hypothesis and Improved R-Estimators of t......Page 527
10.7.3 Asymptotic Bias, MSE Matrix, and Risks of the R-Estimators......Page 528
10.8 R-Estimation of the Parameters with AR[1] Errors......Page 530
10.10 Problems......Page 532
11 Multivariate Models......Page 536
11.1.1 Model, Estimation, and Test of Hypothesis......Page 537
11.1.2 Bias, QB, MSE Matrix, and Weighted Risk Expressions of the Estimators......Page 539
11.1.3 Risk and MSE Analysis of the Estimators......Page 540
11.2 U-statistics Approach to Estimation......Page 543
11.2.1 Asymptotic Properties of Point and Set Estimation under Fixed Alternatives......Page 545
11.2.2 Asymptotic Properties of the Point and Set Estimation under Local Alternatives......Page 546
11.3 Nonparametric Methods: R-estimation......Page 549
11.3.1 Asymptotic Properties of the Point Estimators......Page 551
11.3.2 Asymptotic Properties Confidence Sets......Page 555
11.4.1 Model, Estimation and Tests......Page 557
11.4.2 Preliminary Test and Stein-Type Estimators......Page 558
11.4.3 Bias, Quadratic Bias, MSE Matrices, and Risk Expressions of the Estimators......Page 559
11.4.4 Two-Sample Problem and Estimation of the Means......Page 562
11.4.5 Confidence Sets for the Slope and Intercept Parameters......Page 565
11.5.1 Introduction......Page 566
11.5.2 Asymptotic Properties of the R-estimators......Page 568
11.7 Problems......Page 572
12 Discrete Data Models......Page 576
12.1.1 Model, Estimation, and Test......Page 577
12.1.2 Bayes and Empirical Bayes Estimation......Page 579
12.1.3 Asymptotic Theory of the Estimators and the Test of Departure......Page 581
12.1.4 ADB, ADQB, ADMSE, and ADQR of Estimators......Page 585
12.1.5 Analysis of the Properties of Estimators......Page 586
12.1.6 Baseball Data Analysis......Page 591
12.1.7 Asymptotic Properties of Confidence Sets......Page 594
12.2.2 Model, Estimation, and Test of Hypothesis......Page 596
12.2.3 Asymptotic Theory of the Estimators and the Test-Statistics......Page 599
12.2.4 ADB, ADQB, ADMSE, and ADQR of the Estimators......Page 601
12.2.5 Estimation of Odds Ratio under Uncertain Zero Partial Association......Page 606
12.2.6 Odds Ratios: Application to Meta-analysis of Clinical Trials......Page 607
12.2.3 Revised Estimators of ORs after Deleting OR “Fallis”......Page 611
12.3.3 Test of Independence in an r x c Contingency Table......Page 612
12.3.5 Bayes and Empirical Bayes Method......Page 613
12.3.6 Asymptotic Properties......Page 616
12.3.7 Asymptotic Properties of the Estimators under Local Alternatives......Page 621
12.3.8 Analysis of the Asymptotic Properties of the Estimators......Page 624
12.5 Problems......Page 626
References......Page 628
Glossary......Page 640
Authors Index......Page 642
Subject Index......Page 648
1.1 Display of predicted batting averages based on Stein’s formula......Page 31
3.2.1 Graph of quadratic bias functions of the estimators......Page 89
3.6.1 Graph of AMRE of tPTn and tSn relative to tn......Page 123
3.9.1 Graph of [ps(Ln)]2{E(X4m|Ln)} – 2ps(Ln)E(X2m|Ln)+1......Page 135
4.2.1 Graphs of p|M2(tPTn)|1/p and R2(tPTn; Ip)......Page 158
4.3.1 Geometrical representation of Stein’s idea......Page 165
4.3.2 Graphs of R3(tSn; Ip) and p|M3(tSn)|1/p......Page 169
4.3.3 Graphs of MRE = MRE(tSn; tn) and RRE =RRE(tSn; tn)......Page 171
4.3.4 Graph of QB of estimators: PTE = tPTn, JSE = tSn and PRSE = tS+n......Page 173
4.3.6 Graph of R5(tPT+n; Ip) and R2(tPTn; Ip)......Page 176
4.3.7 Graphs of R4(tS+n; Ip) and R5(tPT+n; Ip)......Page 179
4.4.1 Empirical Bayes Tree......Page 182
4.5.1 Graph of QB of Estimators: PTE, JSE and PRSE......Page 191
4.5.2 Graph of R2(tPTn; s–2Ip), R4(tS+n; s–2Ip), and R5(tPT+n; s–2Ip)......Page 197
4.5.3 Graph of R6(tISn; s–2Ip), R4(tS+n; s–2Ip), and R3(tSn; s–2Ip)......Page 200
12.2.lb Confidence Intervals of odds ratios......Page 609
12.2.2b Confidence Intervals of odds ratios (Deleting Fallis)......Page 610
1.1.1 Batting averages of 18 players......Page 30
3.2.1 Maximum and Minimum Guaranteed Efficiencies for n = 8......Page 92
3.2.2 Maximum and Minimum Guaranteed Efficiencies for n = 12 and x2/Q = 0.1(0.2)0.9......Page 93
3.3.1 Maximum and Minimum Guaranteed Efficiencies......Page 97
3.3.2 Maximum and Minimum Guaranteed Efficiencies......Page 98
3.3.3 Maximum and Minimum Guaranteed Efficiencies......Page 99
3.5.1 Maximum and Minimum Efficiencies of SE and Efficiency of PTE at D0 for Selected a......Page 108
3.5.2 Minimum and Maximum Relative Efficiency of SE and PTE for n = 8, a = .05(.10 ).45 and x2/Q = 1(0.5)5......Page 112
3.6.1 Maximum and Minimum Guaranteed Asymptotic Efficiencies of PTE......Page 124
3.6.2 Maximum and Minimum Guaranteed Asymptotic Efficiencies of PTE......Page 125
4.2.1 Maximum and Minimum Guaranteed MSE Based Efficiencies......Page 161
4.8.1 Decomposition of the Coverage Probability......Page 218
4.8.3 Coverage Probabilities for the Set CPT(tPTn(a)) with g = .10 and b = . 05......Page 221
5.5.1 Maximum and Minimum Guaranteed Efficiencies......Page 263
10.4.2 Empirical Risks for Different Estimators Prior to Testing-Shrinkage Estimates......Page 515
10.4.4 Empirical Risk Values for Shrinkage PTE Based on D–W and G1 Statistic, a = 0.01......Page 516
10.4.6 Empirical Risk Values for Shrinkage PTE Based on D–W and G1 Statistic, a = 0.05......Page 517
10.4.9 Empirical Risk Values for PTE Based on D–W and G1 Statistic, a = 0.01.......Page 518
10.4.11 Empirical Risk Values for PTE Based on D–W and G1 Statistic, a = 0.05.......Page 519
12.1.1 Maximum Relative Efficiencies of the RMLE, PTE, and SE and the Intersection Efficiencies for the PTE and SE for each a with Corresponding A,-Values for pValues for a = 0.05(0.05)0.25 and p = 4(2)16......Page 592
12.1.3 Estimated Average Loss for the Estimators.......Page 593
12.2.2 Various Estimators of Odd-Ratios......Page 608