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Pitman's Measure of Closeness

Pitman's Measure of Closeness
A Comparison of Statistical Estimators

  • Date Published: January 1993
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • format: Paperback
  • isbn: 9780898713084

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About the Authors
  • 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.

    Reviews & endorsements

    '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

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    Product details

    • Date Published: January 1993
    • format: Paperback
    • isbn: 9780898713084
    • length: 242 pages
    • dimensions: 255 x 179 x 14 mm
    • weight: 0.436kg
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • Table of Contents

    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.

  • Authors

    Jerome P. Keating

    Robert L. Mason

    Pranab K. Sen

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