Pitman`s Measure of Closeness – A Comparison of Statistical Estimatiors
A Comparison of Statistical Estimators

This book provides a thorough introduction to the methods and known results associated with PMC.

Jerome P. Keating (Author), Robert L. Mason (Author), Pranab K. Sen (Author)

9780898713084

Paperback / softback, published 31 January 1994

240 pages
25.1 x 17.7 x 1.4 cm, 0.438 kg

' … a comprehensive survey of recent contributions to the subject. It discusses the merits and deficiencies of PMC, throws light on recent controversies, and formulates new problems for further research. Finally, there is a need for such a book, as PMC is not generally discussed in statistical texts. Its role in estimation theory and its usefulness to the decision maker are not well known. . . The contributions by the authors of this book have been especially illuminating in resolving some of the controversies surrounding PMC.' C. R. Rao, from the foreword

Pitman's Measure of Closeness (PMC) is simply an idea whose time has come. Certainly there are many different ways to estimate unknown parameters, but which method should you use? Posed as an alternative to the concept of mean-squared-error, PMC is based on the probabilities of the closeness of competing estimators to an unknown parameter. Renewed interest in PMC over the last 20 years has motivated the authors to produce this book, which explores this method of comparison and its usefulness. Written with research oriented statisticians and mathematicians in mind, but also considering the needs of graduate students in statistics courses, this book provides a thorough introduction to the methods and known results associated with PMC. Following a foreword by C .R. Rao, the first three chapters focus on basic concepts, history, controversy, paradoxes and examples associated with the PMC criterion.

Preface
Part I. Introduction
1. Evolution of Estimation Theory
Least Squares
Method of Moments
Maximum Likelihood
Uniformly Minimum Variance Unbiased Estimation
Biased Estimation
Bayes and Empirical Bayes
Influence Functions and Resampling Techniques
Future Directions
2. PMC Comes of Age
PMC: A Product of Controversy
PMC as an Intuitive Criterion
3. The Scope of the Book
History, Motivation, and Controversy of PMC
A Unified Development of PMC
Part II. Development of Pitman's Measure of Closeness: 1. The Intrinsic Appeal of PMC
Use of MSE
Historical Development of PMC
Convenience Store Example
2. The Concept of Risk
Renyi's Decomposition of Risk
How Do We Understand Risk?
3. Weakness in the Use of Risk
When MSE Does Not Exist
Sensitivity to the Choice of the Loss Function
The Golden Standard
4. Joint Versus Marginal Information
Comparing Estimators with an Absolute Ideal
Comparing Estimators with One Another
5. Concordance of PMC with MSE and MAD
Part III. Anomalies with PMC: 1. Living in an Intransitive World
Round-Robin Competition
Voting Preferences
Transitiveness
2. Paradoxes Among Choice
The Pairwise-Worst Simultaneous-Best Paradox
The Pairwise-Best Simultaneous-Worst Paradox
Politics: The Choice of Extremes
3. Rao's Phenomenom
4. The Question of Ties
Equal Probability of Ties
Correcting the Pitman Criterion
A Randomized Estimator
5. The Rao-Berkson Controversy
Minimum Chi-Square and Maximum Likelihood
Model Inconsistency
Remarks
Part 4. Pairwise Comparisons
1. Geary-Rao Theorem
2. Applications of the Geary-Rao Theorem
3. Karlin's Corollary
4. A Special Case of the Geary-Rao Theorem
Surjective Estimators
The MLR Property
5. Applications of the Special Case
6. Transitiveness
Transitiveness Theorem
Another Extension of Karlin's Corollary
Part V. Pitman-Closest Estimators: 1. Estimation of Location Parameters
2. Estimators of Scale
3. Generalization via Topological Groups
4. Posterior Pitman Closeness
5. Linear Combinations
6. Estimation by Order Statistics
Part 6. Asymptotics and PMC
1. Pitman Closeness of BAN Estimators
Modes of Convergence
Fisher Information
BAN Estimates are Pitman Closet
2. PMC by Asymptotic Representations
A General Proposition
3. Robust Estimation of a Location Parameter
L-Estimators
M-Estimators
R-Estimators
4. APC Characterizations of Other Estimators
Pitman Estimators
Examples of Pitman Estimators
PMC Equivalence
Bayes Estimators
5. Second-Order Efficiency and PMC
Asymptotic Efficiencies
Asymptotic Median Unbiasedness
Higher-Order PMC
Index
Bibliography.

Subject Areas: Miscellaneous items [WZ]