Nonparametric Inference

"This book provides a solid foundation on nonparametric inference for students taking a graduate course in nonparametric statistics and serves as an easily accessible source for researchers in the area." "With the exception of some sections requiring familiarity with measure theory, readers with an advanced calculus background will be comfortable with the material."--Jacket Contents......Page 14 Preface......Page 10 Acknowledgements......Page 12 List of Tables......Page 22 1.1 Sufficient Statistics......Page 24 1.2 Properties of Estimators......Page 25 1.3 Principle of Invariance......Page 26 2.1 Domain of Nonparametric Statistics......Page 30 2.2 Order Statistics......Page 31 2.3 Distribution Theory of Order Statistics......Page 32 2.3.1 Distribution of Sample Range and Mid Range......Page 38 2.3.2 The Distribution of the Median......Page 39 2.3.3 Sampling Distribution of the Coverages......Page 41 2.4 Moments of Order Statistics......Page 43 2.5 Order Statistics: Discrete Populations......Page 53 2.6 Representation of Exponential Order Statistics as a Sum of Independent Random Variables......Page 57 2.7 Representation of General Order Statistics......Page 61 2.8 Angel and Demons' Problems......Page 62 2.9 Large Sample Properties of Order Statistics......Page 66 2.10 Large Sample Properties of Sample Quantiles......Page 68 2.11 Quasi-ranges......Page 77 2.12 Problems......Page 78 3.1 Introduction......Page 81 3.2 Explicit Formulae for Estimators......Page 82 3.3 Estimation for Symmetric Populations......Page 85 3.4 Estimation in a Single Parameter Family......Page 86 3.5 Optimum Properties of Ordered Least Squares Estimates......Page 87 3.6 Examples......Page 91 3.7 Approximations to the Best Linear Estimates......Page 93 3.8 Unbiased Nearly Best Linear Estimates......Page 99 3.9 Nearly Unbiased and Nearly Best Estimates......Page 104 3.10 Inversion of a Useful Matrix......Page 105 3.11 Problems......Page 106 4.1 Confidence Intervals for Quantiles......Page 109 4.2.1 Wilks' (1962) Method......Page 111 4.3 Tolerance Limits......Page 114 4.4 Distribution-free Tolerance Limits......Page 121 4.5 Other Tolerance Limit Problems......Page 124 4.6 Tolerance Regions......Page 125 4.7 Problems......Page 132 5.1 Problems in Non-parametric Estimation......Page 133 5.2 One-sided Con dence Interval for p......Page 140 5.3 Two-sided Confidence Interval for p......Page 145 5.4 Estimation of Distribution Function......Page 147 5.5 Characterization of Distribution-free Statistics......Page 161 5.6 Completeness of the Order Statistic......Page 166 5.7 Problems......Page 172 6.1 Introduction......Page 174 6.2 Difference Quotient Estimate......Page 175 6.3 Class of Estimates of Density Function......Page 177 6.4 Estimate with Prior on Ordinates......Page 185 6.5 Problems......Page 190 7.1 Preliminaries of Hypothesis Testing......Page 191 7.2 Use of Sufficient Statistic......Page 195 7.3 Principle of Invariance......Page 197 7.4 Problems......Page 200 8.2 Chi Square Test......Page 202 8.3 Kolmogorov-Smirnov (K-S) Test......Page 205 8.4 Cram er-von-Mises Test......Page 210 8.5 Shapiro-Wilk (S-W) Test......Page 212 8.6 General Version of S-W Test......Page 217 8.7 Asymptotic Test Based on Spacings......Page 218 8.8 Sherman's Test......Page 219 8.9 Riedwyl Test......Page 220 8.10 Characterization of Distribution-free Tests......Page 221 8.11 Problems......Page 224 9.2 Total Number of Runs......Page 228 9.3 Length of the Longest Run......Page 235 9.4 Runs Up and Down......Page 244 9.5 Runs of Consecutive Elements......Page 250 9.6 Problems......Page 251 10.2 Bivariate Independence......Page 253 10.3 Two-sample Problems......Page 254 10.4 Critical Regions Having Structures......Page 255 10.5 Most Powerful Permutation Tests......Page 257 10.6 One-sample Problems......Page 260 10.7 Tests in Randomized Blocks......Page 261 10.8 Large-sample Power......Page 265 10.9 Modified Permutation Tests......Page 271 10.10 Problems......Page 276 11.2 Correlation between Observations and Ranks......Page 279 11.3 Properties of Rank Orders......Page 283 11.4 Lehmann Alternatives......Page 286 11.5 Two-sample Rank Orders......Page 295 11.6 One-sample Rank Orders......Page 301 11.7 c-sample Rank Orders......Page 306 11.8 Locally Most Powerful (LMP) Rank Tests......Page 311 11.9 Problems......Page 312 12.2 Location Parameter Case......Page 314 12.3 LMP Rank Tests for Scale Changes......Page 317 12.4 Other Tests for Scale Alternatives......Page 320 12.5 Cherno -Savage (CS) Class of Statistics......Page 327 12.6 Problems......Page 331 13.2 LMP Rank Order Test for Location......Page 333 13.4 Tests for Randomness......Page 336 13.5 LMP Rank Tests against Trend......Page 337 13.6 One-sample C-S Class of Statistics......Page 340 13.7 Application to Halperin's Statistic......Page 343 13.8 Problems......Page 346 14.2 Pitman Efficiency......Page 348 14.3 Pitman Efficiency for C-S Class of Statistics......Page 355 14.4.1 Bahadur Efficiency: Limiting Case......Page 357 14.4.2 Bahadur Efficiency: General Setup......Page 362 14.5 Problems......Page 368 15.1 Introduction......Page 369 15.2 LMP Rank Tests......Page 370 15.3 Derivation of the LMP Rank Test......Page 371 15.4 The Variance of the Test Statistic under H0......Page 375 15.5 Other Rank Tests......Page 376 15.6 Variance of Kendall's Test......Page 378 15.7 Asymptotic Normality of a Class of Tests......Page 383 15.8 Tests for Multi-variate Populations......Page 388 15.9 Problems......Page 389 16.2 c-sample Rank Order Tests......Page 391 16.3 Cherno -Savage Class of Statistics......Page 397 16.4 The Median Test......Page 401 16.5 U-Statistics Approach......Page 404 16.6 Combining Two-sample Test Statistics......Page 405 16.7 Kolmogorov-Smirnov Type of Statistics......Page 407 16.8 Problems......Page 409 17.2 Parametric Procedure......Page 411 17.3 Rank Order Tests......Page 418 17.4 A Class of Nonparametric Tests......Page 420 17.5 Problems......Page 422 18.2 Parametric Test Procedures......Page 424 18.3 Nonparametric Test Procedures......Page 433 18.4 Problems......Page 445 19.2 Randomized Block Design......Page 447 19.3 Nonparametric Test Procedures......Page 448 19.4 Nonparametric Tests for Ordered Alternatives......Page 453 19.5 Problems......Page 467 20.1 Introduction......Page 469 20.2 LMP Tests for One-factor Models......Page 471 20.3 Asymptotic Distribution of Logistic Scores Test......Page 475 20.4 Asymptotic Distribution of F-test......Page 484 20.5 Null Distribution and Power Considerations......Page 486 20.6 LMP Tests in Two-way Layouts......Page 487 20.7 LMP Tests in Block Designs......Page 489 20.8 Problems......Page 501 21.1 Introduction and the Model......Page 504 21.3 Certain Remarks......Page 505 21.4 Contrasts in Two-way Layouts......Page 507 21.5 Hodges-Lehmann Type of Estimator......Page 509 21.6 Problems......Page 510 22.2 Brown-Mood Method......Page 513 22.3 Case of a Single Regression Line......Page 514 22.4 Large Sample Approximation......Page 515 22.6 Tests for Regression Parameters......Page 518 22.7 Estimates of Regression Coefficients......Page 521 22.8 Estimates Based on Residuals......Page 526 22.9 Problems......Page 529 23.2 Probability Inequalities......Page 532 23.3.3 Convergence of a Function of Variables......Page 535 23.4 Central Limit Theorems......Page 536 23.5 Dependent Random Variables......Page 537 23.6 Chi-Square for Correlated Variables......Page 539 23.7 Projection Approximations......Page 540 23.8 U-Statistics......Page 544 23.9 Problems......Page 546 24.1 Introduction......Page 550 24.2 Formulation of Problem......Page 551 24.3 Regularity Assumptions......Page 552 24.4 Partition of the Statistic......Page 553 24.5 Alternative Form of the First Order Terms......Page 556 24.6 Scores: Expectations of Order Statistics......Page 559 24.7 Extension to c-sample Case......Page 561 24.8 Dependent Samples Case......Page 564 24.9 Results of H ajek, Pyke and Shorack......Page 565 24.10 Asymptotic Equivalence of Procedures......Page 569 24.11 Problems......Page 573 25.1 Introduction......Page 576 25.2 Regularity Assumptions......Page 577 25.3 Main Theorems......Page 579 25.4 Bounds for Tails and Higher Order Terms......Page 583 25.5 Absolute Normal Scores Test Statistic......Page 585 25.6 Relative Efficiency of Tests for Symmetry......Page 587 25.7 Absolute Normed Scores Test......Page 588 25.8 Application to Halperin's Statistic......Page 591 25.9 c-Sample Case with Random Allocation......Page 594 25.10 Problems......Page 595 26.1 Introduction......Page 597 26.2 Regularity Assumptions......Page 598 26.3 Statement of Main Results......Page 599 26.4 An Application......Page 602 26.6 c-Sample Case......Page 603 26.7 Case of Dependent Samples......Page 605 26.8 Applications......Page 613 26.9 Multivariate Case......Page 617 26.10 Problems......Page 620 27.1 Introduction......Page 622 27.2 Regularity Assumptions......Page 623 27.3 Main Results......Page 624 27.5 c-Sample Case......Page 629 27.6 Applications......Page 631 27.7 Problems......Page 632 Appendix I: Best Estimate of Normal Standard Deviation......Page 634 Appendix II: Confidence Intervals for Median......Page 635 Appendix III: Sample Size for Tolerance Limits......Page 636 Appendix IV: Order Statistics for Tolerance Limits......Page 637 Appendix V: Upper Confidence Bound for P(Y < X)......Page 21 Appendix VI: Confidence Limits for Distribution......Page 639 Bibliography......Page 642 Author Index......Page 676 Subject Index......Page 682 Statistical terminology -- Order statistics -- Ordered least squares estimators -- Interval estimation and tolerance limits -- Nonparametric estimation -- Estimation of density functions -- Review of parametric testing -- Goodness of fit tests -- Randomness tests based on runs -- Permutation tests -- Rank order tests -- LMP tests: Two-sample case -- One-sample rank order tests -- Asymptotic relative efficiency -- LMP tests for independence -- c-sample rank order tests -- c-sample tests for scale -- c-sample tests for ordered alternatives -- Tests in two-way layouts -- Rank tests for random effects -- Estimation of contrasts -- Regression procedures -- Useful asymptotic results -- Asymptotic theory of CS-class of statistics -- CS class for one sample case -- A class of statistics