Pitman's Measure of Closeness

• PrefacePart 1. IntroductionEvolution of Estimation TheoryLeast SquaresMethod of MomentsMaximum LikelihoodUniformly Minimum Variance Unbiased EstimationBiased EstimationBayes and Empirical BayesInfluence Functions and Resampling TechniquesFuture DirectionsChapter 2: PMC Comes of AgePMC: A Product of ControversyPMC as an Intuitive CriterionChapter 3: The Scope of the BookThe History, Motivation, and Controversy of PMCA Unified Development of PMCPart 2. Development of Pitman's Measure of ClosenessChapter 1: The Intrinsic Appeal of PMCUse of MSEHistorical Development of PMCConvenience Store ExampleChapter 2: The Concept of RiskRenyi's Decomposition of RiskHow Do We Understand Risk?Chapter 3: Weakness in the Use of RiskWhen MSE Does Not ExistSensitivity to the Choice of the Loss FunctionThe Golden StandardChapter 4: Joint Versus Marginal InformationComparing Estimators with an Absolute IdealComparing Estimators with One AnotherChapter 5: Concordance of PMC with MSE and MADPart 3. Anomalies with PMCChapter 1: Living in an Intransitive WorldRound-Robin CompetitionVoting PreferencesTransitivenessChapter 2: Paradoxes Among ChoiceThe Pairwise-Worst Simultaneous-Best ParadoxThe Pairwise-Best Simultaneous-Worst ParadoxPolitics: The Choice of ExtremesChapter 3: Rao's PhenomenomChapter 4: The Question of TiesEqual Probability of TiesCorrecting the Pitman CriterionA Randomized EstimatorChapter 5: The Rao-Berkson ControversyMinimum Chi-Square and Maximum LikelihoodModel InconsistencyRemarksPart 4. Pairwise ComparisonsChapter 1: Geary-Rao TheoremChapter 2: Applications of the Geary-Rao TheoremChapter 3: Karlin's CorollaryChapter 4: A Special Case of the Geary-Rao TheoremSurjective EstimatorsThe MLR PropertyChapter 5: Applications of the Special CaseChapter 6: TransitivenessTransitiveness TheoremAnother Extension of Karlin's CorollaryPart 5. Pitman-Closest EstimatorsChapter 1: Estimation of Location ParametersChapter 2: Estimators of ScaleChapter 3: Generalization via Topological GroupsChapter 4: Posterior Pitman ClosenessChapter 5: Linear CombinationsChapter 6: Estimation by Order StatisticsPart 6. Asymptotics and PMCChapter 1: Pitman Closeness of BAN EstimatorsModes of ConvergenceFisher InformationBAN Estimates are Pitman ClosetChapter 2: PMC by Asymptotic RepresentationsA General PropositionChapter 3: Robust Estimation of a Location ParameterL-EstimatorsM-EstimatorsR-EstimatorsChapter 4: APC Characterizations of Other EstimatorsPitman EstimatorsExamples of Pitman EstimatorsPMC EquivalenceBayes EstimatorsChapter 5: Second-Order Efficiency and PMCAsymptotic EfficienciesAsymptotic Median UnbiasednessHigher-Order PMCIndexBibliography.

'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