Developments in Statistics

¿List of ContributorsPrefaceContents of Other VolumesChapter 1 Asymptotic Expansions in Parametric Statistical Theory 1. Introduction 1.1. Purpose of This Chapter 1.2. The General Framework 1.3. Limitations of Nonasymptotic Theory 1.4. Why Are Asymptotic Methods More Promising? 1.5. Is a Refined Asymptotic Theory Really Meaningful? 1.6. How to Read This Chapter 2. Tools of Asymptotic Theory 2.1. The General Problem 2.2. Notations 2.3. Asymptotically Equivalent Sequences of Measures 2.4. Stochastic Expansions 2.5. Edgeworth Expansions 2.6. Edgeworth Expansions for Induced Measures 2.7. Asymptotic Expansions for Integrals 3. Estimation Theory: The General Framework 3.1. Basic Notions 3.2. Notations 3.3. Criteria for Comparing Estimators 3.4. Estimator Sequences Admitting a Stochastic Expansion 4. Estimation Theory Based on Normal Approximations 4.1. Asymptotic Optimality of the Maximum Likelihood Estimator 4.2. An Optimum Property for Asymptotically Median Unbiased Estimator Sequences 4.3. The Order of the Error Term 4.4. Historical Notes 5. The Construction of Asymptotically Efficient Estimators 6. Estimation Theory Based on Approximations of Order o(n-1/2) 6.1. Second-Order Efficiency of Estimator Sequences Admitting a Stochastic Expansion 6.2. The Bias Correction 6.3. Bounds for Estimators without Stochastic Expansion 7. Estimation Theory Based on Approximations of Order o(n-1) 7.1. Outline of the Results 7.2. The Edgeworth Expansion of Order o(n-1) 7.3. A Complete Class of Order o(n-1) 8. Estimating Functions of the Parameter 9. Test Theory for Families without Nuisance Parameters 9.1. Basic Notions 9.2. The Envelope Power Function 9.3. Approximations to the Envelope Power Function of Order o(n0) 9.4. Approximations to the Envelope Power Function of Higher Order 9.5. A Complete Class Theorem 10. Test Theory with Nuisance Parameters: The General Framework 10.1. Basic Notions 10.2. The Envelope Power Function 11. The Construction of Tests of Level a + o(n-8) 11.1. Desensitization 11.2. Studentization 11.3. Tests Obtained from Maximum Likelihood Estimators 12. A Canonical Representation of Stochastic Expansions of Test Statistics 13. The Power of Tests in ??*0 14. The Power of Tests in ??*1 15. The Power of Tests in ??*2 15.1. The Power Function of Order o(n-1) 15.2. An Asymptotically Complete Class of Order o(n-1) 15.3. Concluding Remarks 16. The n-1-Term of the Power Function 16.1. The Problem 16.2. An Example 16.3. Envelope Power Functions 17. The Structure of Deficiencies 17.1. Relative Deficiency of Two Test Sequences 17.2. The Residual Deficiency for Test Sequences 17.3. Deficiency for Estimator Sequences 17.4. The Relationship between Deficiencies for Tests and Estimators 18. Confidence Procedures 18.1. Basic Notions 18.2. Confidence Rays 18.3. Historical Remark 19. Applications of Asymptotic Expansions to Exponential Families 20. On the Numerical Accuracy of Results Based on Edgeworth Expansions ReferencesChapter 2 Orthogonal Models for Contingency Tables 1. Preliminaries 1.1. Introduction 1.2. Definitions 1.3. Combinatorial Methods 1.4. Compatibility of Distributions 1.5. Additive Methods 2. Orthonormal Functions 2.1. Normal Approximations 2.2. General Univariate Distributions 2.3. Special Orthonormal Systems 2.4. Meixner Collection of Random Variables 2.5.