NONPARAMETRIC INFERENCE
    Z GOVINDARAJUJU
University of Kentucky, USA
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1 Statistical T erminology 1
1.1 Sucient Statistics . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Properties of Estimators . . . . . . . . . . . . . . . . . . . 2
1.3 Principle of Inv ariance . . . . . . . . . . . . . . . . . . . . 3
2 Order Statistics 7
2.1 Domain of Nonparametric Statistics . . . . . . . . . . . . . 7
2.2 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Distribution Theory of Order Statistics . . . . . . . . . . . 9
2.3.1 Distribution of Sample Range and Mid Range . . . 15
2.3.2 The Distribution of the Median . . . . . . . . . . . 16
2.3.3 Sampling Distribution of the Coverages . . . . . . . 18
2.4 Moments of Order Statistics . . . . . . . . . . . . . . . . . 20
2.5 Order Statistics: Discrete Populations . . . . . . . . . . . . 30
2.6 Representation of Exponential Order Statistics as a Sum
of Independent Random V ariables . . . . . . . . . . . . . . 34
2.7 Representation of General Order Statistics . . . . . . . . . 38
2.8 Angel and Demons' Problems . . . . . . . . . . . . . . . . 39
2.9 Large Sample Properties of Order Statistics . . . . . . . . 43
2.10 Large Sample Properties of Sample Quantiles . . . . . . . 45
2.11 Quasi-ranges . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Ordered Least Squares Estimators 58
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Explicit F ormulae for Estimators . . . . . . . . . . . . . . . 59
3.3 Estimation for Symmetric Populations . . . . . . . . . . . 62
3.4 Estimation in a Single Parameter F amily . . . . . . . . . . 63
3.5 Optimum Properties of Ordered Least Squares Estimates . 64
3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.7 Approximations to the Best Linear Estimates . . . . . . . 70
3.8 Unbiased Nearly Best Linear Estimates . . . . . . . . . . . 76
3.9 Nearly Unbiased and Nearly Best Estimates . . . . . . . . 81
3.10 Inversion of a Useful Matrix . . . . . . . . . . . . . . . . . 82
3.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Interv al Estimation and T olerance Limits 86
4.1 Condence Interv als for Quantiles . . . . . . . . . . . . . . 86
4.2 Large Sample Condence Interv als . . . . . . . . . . . . . . 88
4.2.1 Wilks' (1962) Method . . . . . . . . . . . . . . . . . 88
4.3 T olerance Limits . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4 Distribution-free T olerance Limits . . . . . . . . . . . . . . 98
4.5 Other T olerance Limit Problems . . . . . . . . . . . . . . . 101
4.6 T olerance Regions . . . . . . . . . . . . . . . . . . . . . . . 102
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Nonparametric Estimation 110
5.1 Problems in Non-parametric Estimation . . . . . . . . . . 110
5.2 One-sided Condence Interv al for p . . . . . . . . . . . . . 117
5.3 Two-sided Condence Interv al for p . . . . . . . . . . . . . 122
5.4 Estimation of Distribution F unction . . . . . . . . . . . . . 124
5.5 Characterization of Distribution-free Statistics . . . . . . . 138
5.6 Completeness of the Order Statistic . . . . . . . . . . . . . 143
5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6 Estimation of Density F unctions 151
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Dierence Quotient Estimate . . . . . . . . . . . . . . . . . 152
6.3 Class of Estimates of Density F unction . . . . . . . . . . . 154
6.4 Estimate with Prior on Ordinates . . . . . . . . . . . . . . 162
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7 Review of Parametric T esting 168
8 Goodness of Fit T ests 179
9 Randomness T ests Based on Runs 205
10 Permutation T ests 230
11 Rank Order T ests 256
12 LMP T ests: Two-sample Case 291
13 One-sample Rank Order T ests 310
14 Asymptotic Relative Eciency 325
15 LMP T ests for Independence 346
16 c-sample Rank Order T ests 368
17 c-sample T ests for Scale 388
18 c-sample T ests for Ordered Alternatives 401
19 T ests in Two-way Layouts 424
20 Rank T ests for Random Eects 446
21 Estimation of Contrasts 481
22 Regression Procedures 490
23 Useful Asymptotic Results 509
24 Asymptotic Theory of CS-class of Statistics 527
25 CS Class for One Sample Case 553
26 A Class of Statistics 574
27 Systematic Statistics 599

Appendices 611
Appendix I: Best Estimate of Normal Standard Deviation . . . . 611
Appendix II: Condence Interv als for Median . . . . . . . . . . . 612
Appendix III: Sample Size for T olerance Limits . . . . . . . . . . 613
Appendix IV: Order Statistics for T olerance Limits . . . . . . . . 614
Appendix V: Upper Condence Bound for P (Y < X) . . . . . . . 615
Appendix VI: Condence Limits for Distribution . . . . . . . . . 616