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. The material is illustrated through realistic estimation problems and presented with a limited degree of technical difficulty. The last three chapters present a unified development of the extensive theoretical and mathematical research on PMC, albeit in the setup of a single parameter. Taken together, they serve as a single comprehensive source on this important topic. The text is highly referenced, allowing researchers to readily access the original articles. What lies ahead for PMC? This book begins to answer that question by presenting a unified discourse on this alternative criterion to traditional methods of comparison. Find out how the authors have begun to unravel the many attributes of PMC and their connections to other estimation criteria.
Preface Part 1. Introduction 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 Chapter 2: PMC Comes of Age PMC: A Product of Controversy PMC as an Intuitive Criterion Chapter 3: The Scope of the Book The History, Motivation, and Controversy of PMC A Unified Development of PMC Part 2. Development of Pitman's Measure of Closeness Chapter 1: The Intrinsic Appeal of PMC Use of MSE Historical Development of PMC Convenience Store Example Chapter 2: The Concept of Risk Renyi's Decomposition of Risk How Do We Understand Risk? Chapter 3: Weakness in the Use of Risk When MSE Does Not Exist Sensitivity to the Choice of the Loss Function The Golden Standard Chapter 4: Joint Versus Marginal Information Comparing Estimators with an Absolute Ideal Comparing Estimators with One Another Chapter 5: Concordance of PMC with MSE and MAD Part 3. Anomalies with PMC Chapter 1: Living in an Intransitive World Round-Robin Competition Voting Preferences Transitiveness Chapter 2: Paradoxes Among Choice The Pairwise-Worst Simultaneous-Best Paradox The Pairwise-Best Simultaneous-Worst Paradox Politics: The Choice of Extremes Chapter 3: Rao's Phenomenom Chapter 4: The Question of Ties Equal Probability of Ties Correcting the Pitman Criterion A Randomized Estimator Chapter 5: The Rao-Berkson Controversy Minimum Chi-Square and Maximum Likelihood Model Inconsistency Remarks Part 4. Pairwise Comparisons Chapter 1: Geary-Rao Theorem Chapter 2: Applications of the Geary-Rao Theorem Chapter 3: Karlin's Corollary Chapter 4: A Special Case of the Geary-Rao Theorem Surjective Estimators The MLR Property Chapter 5: Applications of the Special Case Chapter 6: Transitiveness Transitiveness Theorem Another Extension of Karlin's Corollary Part 5. Pitman-Closest Estimators Chapter 1: Estimation of Location Parameters Chapter 2: Estimators of Scale Chapter 3: Generalization via Topological Groups Chapter 4: Posterior Pitman Closeness Chapter 5: Linear Combinations Chapter 6: Estimation by Order Statistics Part 6. Asymptotics and PMC Chapter 1: Pitman Closeness of BAN Estimators Modes of Convergence Fisher Information BAN Estimates are Pitman Closet Chapter 2: PMC by Asymptotic Representations A General Proposition Chapter 3: Robust Estimation of a Location Parameter L-Estimators M-Estimators R-Estimators Chapter 4: APC Characterizations of Other Estimators Pitman Estimators Examples of Pitman Estimators PMC Equivalence Bayes Estimators Chapter 5: Second-Order Efficiency and PMC Asymptotic Efficiencies Asymptotic Median Unbiasedness Higher-Order PMC Index Bibliography.
'This monograph was written by three internationally reputed statisticians who have been instrumental in creating the recent interest in Pitman's measure of closeness (PMC)... There are several motivating and nontechnical examples that are easy to follow. Some material from Chapters 1, 2, and 3 could be used in a first-year graduate level course, whereas Chapters 4, 5, and 6 could be discussed in a graduate level course. Implicitly, in addition to providing answers to some difficult questions, the authors pose many important and challenging research problems for future research... I believe that this is a useful monograph for a researcher interested in the PMC criterion. It summarizes and unifies some of the important results in the area.' Shyamal D. Peddada, Journal of Applied Statistical Applications 'The authors have written an interesting and lively account of recent developments in the study of Pitman Closeness. The book gathers together much of what is known in the area and presents it in a balanced manner. It is the best and most complete source of material on Pitman's measure of closeness and should be most useful to anyone interested in the subject.' William E. Strawderman, Professor of Statistics, Rutgers University 'Nicely presents history of Pitman's measure of closeness (PMC), applications to single-parameter estimation problems, PMC anomalies, and asymtotics.' American Mathematical Monthly 'This recent monograph assembles the widespread material concerning Pitman's measure of closeness (PMC) that is available in the literature and much of which is not widely known ... the authors recommend Pitman closeness as an interesting alternative criterion for comparing estimators. They investigate the usefulness of the PMC for this purpose, and discuss the properties of 'Pitman-closest' estimators. This is done both from a frequentist and Bayesian point of view, in both small-sample and large-sample settings. The book contains many fascinating examples and results.' E. L. Lehmann, Short Book Reviews ' ... 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 "A holistic presentation of the known results about PMC, presented under a common notation, with the aim of accelerating the integration of the beneficial features of this criterion into the mainstream of statistical through. Written for theorists, practicioners, and students -- at a level appropriate for graduate students who have completed a traditional two-semester course in mathematical statistics." -- SciTech Book News, June 1993
