# Developments in Statistics, Volume 3

## 1st Edition

**Editor:**Paruchuri R. Krishnaiah

**eBook ISBN:**9781483264202

**Imprint:**Academic Press

**Published Date:**28th November 1980

**Page Count:**268

**View all volumes in this series:**Developments in Statistics

## Table of Contents

List of Contributors

Preface

Contents of Other Volumes

Chapter 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 α + 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 𝒮*0
14. The Power of Tests in 𝒮*1

15. The Power of Tests in 𝒮*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

References

Chapter 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. Relations between Orthonormal Systems

2.6. Choice of Intervals for the Pearson χ2 Test

2.7. Choice of Test Criterion

3. Models for Bivariate Distributions

3.1. Orthogonal Properties

3.2. Biorthogonal Properties

3.3. Generalities on Regression

3.4. Convolutions of Meixner Distributions

3.5. Biorthogonal Meixner Distributions

3.6. Miscellaneous Meixner Distributions

3.7. Pearson Models and Stochastic Processes

4. Models for Multivariate Distributions

4.1. Orthogonal Properties

4.2. Mutual Independence in Several Dimensions

4.3. Multivariate Meixner Distributions

5. Applications

5.1. Tests of Independence

5.2. Tests of Goodness of Fit

5.3. Tests Based on the Maximal Correlation

5.4. Tests of Biorthogonality

5.5. Miscellaneous Applications

References

Chapter 3 The Increased Use of Statistical Concepts in Economic Analysis

1. Introduction

2. Concepts from Statistical Information Theory

2.1. The Measurement of Income Inequality

2.2. Intergenerational Occupational Mobility

2.3. Other Applications of Informational Concepts

3. Moments of Univariate and Bivariate Distributions

3.1. Divisia Moments

3.2. The True Indexes of Real Income and the Cost of Living

3.3. Frisch Moments and the True Marginal Price Index

3.4. Extensions

4. A Statistical View of Microeconomic Theory

4.1. The Differential Approach to Consumption Theory

4.2. Frisch Moments and True Indexes Further Considered

4.3. Block Structures

4.4. Finite-Change Formulations

4.5. The Measurement of the Quality of Consumption

4.6. Concluding Remarks

5. The Theory of Rational Random Behavior

5.1. The Exact Theory

5.2. The Asymptotic Theory

5.3. Application to Consumer Demand

5.4. Concluding Remarks

6. A Constrained Principal Component Transformation

6.1. The Preference Independence Transformation

6.2. Derivations and Further Results

6.3. Application to the Demand for Meat

6.4. A Principal Component Interpretation

6.5. Concluding Remarks

7. Conclusion

References

Chapter 4 Path Analysis: An Exposition

1. Introduction

2. Predictor Regression

3. Path or Linear Regression Causal Analysis

4. Internal Consistency and Overidentification

5. Matrix Formulations, Recursiveness, and “Koopmans-Hood” Assumptions

6. Decomposition of Correlation

7. Other Topics, in Brief

7.1. Statistical Inference

7.2 Systems of Special Form

7.3. Some Bibliographic Notes

Appendix: Proofs

References

Author Index

Subject Index

## Description

Development in Statistics, Volume 3 is a collection of papers that deals with asymptotic expansions in parametric statistical theory, orthogonal models for contingency tables, statistical concepts in economic analysis, and an exposition of path analysis. One paper presents an inference model based on a sample of independent identically distributed observations to arrive at a general statistical theory founded on asymptotic methods. Another paper discusses the applicability of statistical concepts to economics and related areas, with emphasis on not-so-obvious applications (known as utility and expected loss). The paper explains information theory concepts for the measurement of income inequality, intergenerational occupational mobility, as well as to first- and second-order moments of univariate and bivariate distributions (such as measurements applied to the cost of living and of real income). One paper notes that the starting point in path analysis is a linear predictor (in the least-squares sense) for one random variable in terms of a number of others. The paper adds that the work of Koopmans and Hood (1953) on econometrics is part of the starting point. Statisticians, economists, mathematicians, students, and professors of calculus or advanced mathematics will surely appreciate the collection.

## Details

- No. of pages:
- 268

- Language:
- English

- Copyright:
- © Academic Press 1980

- Published:
- 28th November 1980

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483264202