Concepts of Nonparametric Theory

'1 Concepts of Statistical Inference and the Binomial Distribution.- 1 Introduction.- 2 Probability Distributions.- 3 Estimators and their Properties.- 3.1 Unbiasedness and Variance.- 3.2 Consistency.- 3.3 Sufficiency.- 3.4 Minimum Variance.- 4 Hypothesis Testing.- 4.1 Tests and their Interpretation.- 4.2 Errors.- 4.3 One-Tailed Binomial Tests.- P-values.- 4.5 Two-Tailed Test Procedures and P-values.- 4.6 Other Conclusions in Two-Tailed Tests.- 5 Randomized Test Procedures.- 5.1 Introduction: Motivation and Examples.- 5.2 Randomized Tests: Definitions.- 5.3 Nonrandomized Tests Equivalent to Randomized Tests.- 5.4 Usefulness of Randomized Tests in Theory and Practice.- 5.5 ?Randomized P-values.- 6 Confidence Regions.- 6.1 Definition and Construction in the Binomial Case.- 6.2 Definition of Confidence Regions and Relationship to Tests in the General Case.- 6.3 Interpretation of Confidence Regions.- 6.4 True Confidence Level.- 6.5 Including False Values and the Size of Confidence Regions.- 6.6 ?Randomized Confidence Regions.- 7 Properties of One-Tailed Procedures.- 7.1 Uniformly Most Powerful One-Tailed Tests.- 7.2 ?Admissibility and Completeness of One-Tailed Tests.- 7.3 Confidence Procedures.- 7.4 ?Proofs.- 8 Choice of Two-Tailed Procedures and their Properties.- 8.1 Test Procedures.- 8.2 Confidence Procedures.- 8.3 ?Completeness and Admissibility of Two-Conclusion Two-Tailed Tests.- 8.4 ?Completeness and Admissibility of Three-Conclusion Two-Tailed Tests.- 9 Appendices to Chapter 1.- A Limits of the Binomial Distribution.- A.1 Ordinary Poisson Approximation and Limit.- A.2 Ordinary Normal Approximation and Limit.- ? Convergence in Distribution and Asymptotic Distributions.- B.1 Convergence of Frequency Functions and Densities.- B.2 Convergence in Distribution.- B.3 Two Central Limit Theorems.- Problems.- 2 One-Sample and Paired-Sample Inferences Based on the Binomial Distribution.- 1 Introduction.- 2 Quantile Values.- 3 The One-Sample Sign Test for Quantile Values.- 3.1 Test Procedures.- 3.2 "Optimum" Properties.- 3.3 ?Proofs.- 4 Confidence Procedures Based on the Sign Test.- 5 Interpolation between Attainable Levels.- 6 The Sign Test with Zero Differences.- 6.1 Discussion of Procedures.- 6.2 Conditional Properties of Conditional Sign Tests.- 6.3 Unconditional Properties of Conditional Sign Tests.- 6.4 ?Proof for One-Sided Alternatives.- 6.5 ?Prooffor Two-Sided Alternatives.- 7 Paired Observations.- 8 Comparing Proportions using Paired Observations.- 8.1 Test Procedure.- 8.2 Alternative Presentations.- 8.3 Example.- 8.4 Interpretation of the Test Results.- 8.5 Properties of the Test.- 8.6 Other Inferences.- 8.7 Cox Model.- 9 Tolerance Regions.- 9.1 Definition.- 9.2 Practical Uses.- 9.3 Construction of Tolerance Regions: Wilks' Method.- 9.4 Tolerance Regions for Description.- 9.5 Tolerance Regions for Prediction.- 9.6 More General Construction Procedures.- Problems.- 3 One-Sample and Paired-Sample Inferences Based on Signed Ranks.- 1 Introduction.- 2 The Symmetry Assumption or Hypothesis.- 3 The Wilcoxon Signed-Rank Test.- 3.1 Test Procedure and Exact Null Distribution Theory.- 3.2 Asymptotic Null Distribution Theory.- 3.3 Large Sample Power.- 3.4 Consistency.- 3.5 Weakening the Assumptions.- 4 Confidence Procedures Based on the Wilcoxon Signed-Rank Test.- 5 A Modified Wilcoxon Procedure.- 6 Zeros and Ties.- 6.1 Introduction.- 6.2 Obtaining the Signed Ranks.- 6.3 Test Procedures.- 6.4 Warnings and Anomalies: Examples.- 6.5 Comparison of Procedures.- 7 Other Signed-Rank Procedures.- 7.1 Sums of Signed Constants.- 7.2 Signed Ranks and Walsh Averages.- 7.3 Confidence Bounds Corresponding to Signed-Rank Tests.- 7.4 Procedures Involving a Small Number of Walsh Averages.- 8 Invariance and Signed-Rank Procedures.- 8.1 Permutation Invariance.- 8.2 Invariance under Increasing, Odd Transformations.- 9 Locally Most Powerful Signed-Rank Tests