Nonparametric Statistical Methods

Preface xiii

1. Introduction 1

1.1. Advantages of Nonparametric Methods 1

1.2. The Distribution-Free Property 2

1.3. Some Real-World Applications 3

1.4. Format and Organization 6

1.5. Computing with R 8

1.6. Historical Background 9

2. The Dichotomous Data Problem 11

Introduction 11

2.1. A Binomial Test 11

2.2. An Estimator for the Probability of Success 22

2.3. A Confidence Interval for the Probability of Success (Wilson) 24

2.4. Bayes Estimators for the Probability of Success 33

3. The One-Sample Location Problem 39

Introduction 39

Paired Replicates Analyses by Way of Signed Ranks 39

3.1. A Distribution-Free Signed Rank Test (Wilcoxon) 40

3.2. An Estimator Associated with Wilcoxon's Signed Rank Statistic (Hodges-Lehmann) 56

3.3. A Distribution-Free Confidence Interval Based on Wilcoxon's Signed Rank Test (Tukey) 59

Paired Replicates Analyses by Way of Signs 63

3.4. A Distribution-Free Sign Test (Fisher) 63

3.5. An Estimator Associated with the Sign Statistic (Hodges-Lehmann) 76

3.6. A Distribution-Free Confidence Interval Based on the Sign Test (Thompson, Savur) 80

One-Sample Data 84

3.7. Procedures Based on the Signed Rank Statistic 84

3.8. Procedures Based on the Sign Statistic 90

3.9. An Asymptotically Distribution-Free Test of Symmetry (Randles-Fligner-Policello-Wolfe, Davis-Quade) 94

Bivariate Data 102

3.10. A Distribution-Free Test for Bivariate Symmetry (Hollander) 102

3.11. Efficiencies of Paired Replicates and One-Sample Location Procedures 112

4. The Two-Sample Location Problem 115

Introduction 115

4.1. A Distribution-Free Rank Sum Test (Wilcoxon, Mann and Whitney) 115

4.2. An Estimator Associated with Wilcoxon's Rank Sum Statistic (Hodges-Lehmann) 136

4.3. A Distribution-Free Confidence Interval Based on Wilcoxon's Rank Sum Test (Moses) 142

4.4. A Robust Rank Test for the Behrens-Fisher Problem (Fligner-Policello) 145

4.5. Efficiencies of Two-Sample Location Procedures 149

5. The Two-Sample Dispersion Problem and Other Two-Sample Problems 151

Introduction 151

5.1. A Distribution-Free Rank Test for Dispersion-Medians Equal (Ansari-Bradley) 152

5.2. An Asymptotically Distribution-Free Test for Dispersion Based on the Jackknife-Medians Not Necessarily Equal (Miller) 169

5.3. A Distribution-Free Rank Test for Either Location or Dispersion (Lepage) 181

5.4. A Distribution-Free Test for General Differences in Two Populations (Kolmogorov-Smirnov) 190

5.5. Efficiencies of Two-Sample Dispersion and Broad Alternatives Procedures 200

6. The One-Way Layout 202

Introduction 202

6.1. A Distribution-Free Test for General Alternatives (Kruskal-Wallis) 204

6.2. A Distribution-Free Test for Ordered Alternatives (Jonckheere-Terpstra) 215

6.3. Distribution-Free Tests for Umbrella Alternatives (Mack-Wolfe) 225

6.3A. A Distribution-Free Test for Umbrella Alternatives, Peak Known (Mack-Wolfe) 226

6.3B. A Distribution-Free Test for Umbrella Alternatives, Peak Unknown (Mack-Wolfe) 241

6.4. A Distribution-Free Test for Treatments Versus a Control (Fligner-Wolfe) 249

Rationale For Multiple Comparison Procedures 255

6.5. Distribution-Free Two-Sided All-Treatments Multiple Comparisons Based on Pairwise Rankings-General Configuration (Dwass, Steel, and Critchlow-Fligner) 256

6.6. Distribution-Free One-Sided All-Treatments Multiple Comparisons Based on Pairwise Rankings-Ordered Treatment Effects (Hayter-Stone) 265

6.7. Distribution-Free One-Sided Treatments-Versus-Control Multiple Comparisons Based on Joint Rankings (Nemenyi, Damico-Wolfe) 271

6.8. Contrast Estimation Based on Hodges-Lehmann Two-Sample Estimators (Spjøtvoll) 278

6.9. Simultaneous Confidence Intervals for All Simple Contrasts (Critchlow-Fligner) 282

6.10. Efficiencies of One-Way Layout Procedures 287

7. The Two-Way Layout 289

Introduction 289

7.1. A Distribution-Free Test for General Alternatives in a Randomized Complete Block Design (Friedman, Kendall-Babington Smith) 292

7.2. A Distribution-Free Test for Ordered Alternatives in a Randomized Complete Block Design (Page) 304

Rationale for Multiple Comparison Procedures 315

7.3. Distribution-Free Two-Sided All-Treatments Multiple Comparisons Based on Friedman Rank Sums-General Configuration (Wilcoxon, Nemenyi, McDonald-Thompson) 316

7.4. Distribution-Free One-Sided Treatments Versus Control Multiple Comparisons Based on Friedman Rank Sums (Nemenyi, Wilcoxon-Wilcox, Miller) 322

7.5. Contrast Estimation Based on One-Sample Median Estimators (Doksum) 328

Incomplete Block Data-Two-Way Layout with Zero or One Observation Per Treatment-Block Combination 331

7.6. A Distribution-Free Test for General Alternatives in a Randomized Balanced Incomplete Block Design (BIBD) (Durbin-Skillings-Mack) 332

7.7. Asymptotically Distribution-Free Two-Sided All-Treatments Multiple Comparisons for Balanced Incomplete Block Designs (Skillings-Mack) 341

7.8. A Distribution-Free Test for General Alternatives for Data From an Arbitrary Incomplete Block Design (Skillings-Mack) 343

Replications-Two-Way Layout with at Least One Observation for Every Treatment-Block Combination 354

>1) of Replications Per Treatment-Block Combination (Mack-Skillings) 354

7.10. Asymptotically Distribution-Free Two-Sided All-Treatments Multiple Comparisons for a Two-Way Layout with an Equal Number of Replications in Each Treatment-Block Combination (Mack-Skillings) 367

Analyses Associated with Signed Ranks 370

7.11. A Test Based on Wilcoxon Signed Ranks for General Alternatives in a Randomized Complete Block Design (Doksum) 370

7.12. A Test Based on Wilcoxon Signed Ranks for Ordered Alternatives in a Randomized Complete Block Design (Hollander) 376

7.13. Approximate Two-Sided All-Treatments Multiple Comparisons Based on Signed Ranks (Nemenyi) 379

7.14. Approximate One-Sided Treatments-Versus-Control Multiple Comparisons Based on Signed Ranks (Hollander) 382

7.15. Contrast Estimation Based on the One-Sample Hodges-Lehmann Estimators (Lehmann) 386

7.16. Efficiencies of Two-Way Layout Procedures 390

8. The Independence Problem 393

Introduction 393

8.1. A Distribution-Free Test for Independence Based on Signs (Kendall) 393

8.2. An Estimator Associated with the Kendall Statistic (Kendall) 413

8.3. An Asymptotically Distribution-Free Confidence Interval Based on the Kendall Statistic (Samara-Randles, Fligner-Rust, Noether) 415

8.4. An Asymptotically Distribution-Free Confidence Interval Based on Efron's Bootstrap 420

8.5. A Distribution-Free Test for Independence Based on Ranks (Spearman) 427

8.6. A Distribution-Free Test for Independence Against Broad Alternatives (Hoeffding) 442

8.7. Efficiencies of Independence Procedures 450

9. Regression Problems 451

Introduction 451

One Regression Line 452

9.1. A Distribution-Free Test for the Slope of the Regression Line (Theil) 452

9.2. A Slope Estimator Associated with the Theil Statistic (Theil) 458

9.3. A Distribution-Free Confidence Interval Associated with the Theil Test (Theil) 460

9.4. An Intercept Estimator Associated with the Theil Statistic and Use of the Estimated Linear Relationship for Prediction (Hettmansperger-McKean-Sheather) 463

k(=2) Regression Lines 466

9.5. An Asymptotically Distribution-Free Test for the Parallelism of Several Regression Lines (Sen, Adichie) 466

General Multiple Linear Regression 475

9.6. Asymptotically Distribution-Free Rank-Based Tests for General Multiple Linear Regression (Jaeckel, Hettmansperger-McKean) 475

Nonparametric Regression Analysis 490

9.7. An Introduction to Non-Rank-Based Approaches to Nonparametric Regression Analysis 490

9.8. Efficiencies of Regression Procedures 494

10. Comparing Two Success Probabilities 495

Introduction 495

10.1. Approximate Tests and Confidence Intervals for the Difference between Two Success Probabilities (Pearson) 496

10.2. An Exact Test for the Difference between Two Success Probabilities (Fisher) 511

10.3. Inference for the Odds Ratio (Fisher, Cornfield) 515

10.4. Inference for k Strata of 2 × 2 Tables (Mantel and Haenszel) 522

10.5. Efficiencies 534

11. Life Distributions and Survival Analysis 535

Introduction 535

11.1. A Test of Exponentiality Versus IFR Alternatives (Epstein) 536

11.2. A Test of Exponentiality Versus NBU Alternatives (Hollander-Proschan) 545

11.3. A Test of Exponentiality Versus DMRL Alternatives (Hollander-Proschan) 555

11.4. A Test of Exponentiality Versus a Trend Change in Mean Residual Life (Guess-Hollander-Proschan) 563

11.5. A Confidence Band for the Distribution Function (Kolmogorov) 568

11.6. An Estimator of the Distribution Function When the Data are Censored (Kaplan-Meier) 578

11.7. A Two-Sample Test for Censored Data (Mantel) 594

11.8. Efficiencies 605

12. Density Estimation 609

Introduction 609

12.1. Density Functions and Histograms 609

12.2. Kernel Density Estimation 617

12.3. Bandwidth Selection 624

12.4. Other Methods 628

13. Wavelets 629

Introduction 629

13.1. Wavelet Representation of a Function 630

13.2. Wavelet Thresholding 644

13.3. Other Uses of Wavelets in Statistics 655

14. Smoothing 656

Introduction 656

14.1. Local Averaging (Friedman) 657

14.2. Local Regression (Cleveland) 662

14.3. Kernel Smoothing 667

14.4. Other Methods of Smoothing 675

15. Ranked Set Sampling 676

Introduction 676

15.1. Rationale and Historical Development 676

15.2. Collecting a Ranked Set Sample 677

15.3. Ranked Set Sampling Estimation of a Population Mean 685

15.4. Ranked Set Sample Analogs of the Mann-Whitney-Wilcoxon Two-Sample Procedures (Bohn-Wolfe) 717

15.5. Other Important Issues for Ranked Set Sampling 737

15.6. Extensions and Related Approaches 742

16. An Introduction to Bayesian Nonparametric Statistics via the Dirichlet Process 744

Introduction 744

16.1. Ferguson's Dirichlet Process 745

16.2. A Bayes Estimator of the Distribution Function (Ferguson) 749

16.3. Rank Order Estimation (Campbell and Hollander) 752

16.4. A Bayes Estimator of the Distribution When the Data are Right-Censored (Susarla and Van Ryzin) 755

16.5. Other Bayesian Approaches 759

Bibliography 763

R Program Index 791

Author Index 799

Subject Index 809