Nonparametric inference

Z. Govindarajulu9789812700346, 9789812770400, 981270034X

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.

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Table of contents :
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