Pitman's Measure of Closeness: A Comparison of Statistical Estimatorsby Robert L. Mason, P. K. Sen, Pranab K. Sen
Pub. Date: 01/28/1993
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.
Table of ContentsPreface; 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.
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