Pitman's Measure of Closeness

by Keating, Jerome P.; Mason, Robert Lee; Sen, Pranab Kumar

• ISBN13: 9780898713084
• ISBN10: 0898713080
• Format: Paperback
• Copyright: 1993-03-01
• Publisher: Society for Industrial & Applied

Summary

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 Contents

Preface

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

PMC Comes of Age

PMC: A Product of Controversy

PMC as an Intuitive Criterion

The Scope of the Book

History, Motivation, and Controversy of PMC

A Unified Development of PMC

Development of Pitman's Measure of Closeness:

The Intrinsic Appeal of PMC

Use of MSE

Historical Development of PMC

Convenience Store Example

The Concept of Risk

Renyi's Decomposition of Risk

How Do We Understand Risk?

Weakness in the Use of Risk

When MSE Does Not Exist

Sensitivity to the Choice of the Loss Function

The Golden Standard

Joint Versus Marginal Information

Comparing Estimators with an Absolute Ideal

Comparing Estimators with One Another

Concordance of PMC with MSE and MAD

Anomalies with PMC:

Living in an Intransitive World

Round-Robin Competition

Voting Preferences

Transitiveness

Paradoxes Among Choice

The Pairwise-Worst Simultaneous-Best Paradox

The Pairwise-Best Simultaneous-Worst Paradox

Politics: The Choice of Extremes

Rao's Phenomenom

The Question of Ties

Equal Probability of Ties

Correcting the Pitman Criterion

A Randomized Estimator

The Rao-Berkson Controversy

Minimum Chi-Square and Maximum Likelihood

Model Inconsistency

Remarks

Pairwise Comparisons

Geary-Rao Theorem

Applications of the Geary-Rao Theorem

Karlin's Corollary

A Special Case of the Geary-Rao Theorem

Surjective Estimators

The MLR Property

Applications of the Special Case

Transitiveness

Transitiveness Theorem

Another Extension of Karlin's Corollary

Pitman-Closest Estimators:

Estimation of Location Parameters

Estimators of Scale

Generalization via Topological Groups

Posterior Pitman Closeness

Linear Combinations

Estimation by Order Statistics

Asymptotics and PMC

Pitman Closeness of BAN Estimators

Modes of Convergence

Fisher Information

BAN Estimates are Pitman Closet

PMC by Asymptotic Representations

A General Proposition

Robust Estimation of a Location Parameter

L-Estimators

M-Estimators

R-Estimators

APC Characterizations of Other Estimators

Pitman Estimators

Examples of Pitman Estimators

PMC Equivalence

Bayes Estimators

Second-Order Efficiency and PMC

Asymptotic Efficiencies

Asymptotic Median Unbiasedness

Higher-Order PMC

Index

Bibliography

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