This book explores both non parametric and general statistical ideas by developing non parametric procedures in simple situations. The major goal is to give the reader a thorough intuitive understanding of the concepts underlying nonparametric procedures and a full appreciation of the properties and operating characteristics of those procedures covered. This book differs from most statistics books by including considerable philosophical and methodological discussion. Special attention is given to discussion of the strengths and weaknesses of various statistical methods and approaches.…mehr
This book explores both non parametric and general statistical ideas by developing non parametric procedures in simple situations. The major goal is to give the reader a thorough intuitive understanding of the concepts underlying nonparametric procedures and a full appreciation of the properties and operating characteristics of those procedures covered. This book differs from most statistics books by including considerable philosophical and methodological discussion. Special attention is given to discussion of the strengths and weaknesses of various statistical methods and approaches. Difficulties that often arise in applying statistical theory to real data also receive substantial attention. The approach throughout is more conceptual than mathematical. The "Theorem-Proof" format is avoided; generally, properties are "shown," rather than "proved." In most cases the ideas behind the proof of an im portant result are discussed intuitively in the text and formal details are left as an exercise for the reader. We feel that the reader will learn more from working such things out than from checking step-by-step a complete presen tation of all details.
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Inhaltsangabe
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 181 Problems.- 4 One-Sample and Paired-Sample Inferences Based on the Method of Randomization.- 1
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 181 Problems.- 4 One-Sample and Paired-Sample Inferences Based on the Method of Randomization.- 1
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