Samaniego F.J. A Comparison of the Bayesian and Frequentist Approaches to Estimation

Springer – 2010, 226 pages
ISBN: 1441959408, 9781441959416This monograph contributes to the area of comparative statistical inference. Attention is restricted to the important subfield of statistical estimation. The book is intended for an audience having a solid grounding in probability and statistics at the level of the year-long undergraduate course taken by statistics and mathematics majors. The necessary background on Decision Theory and the frequentist and Bayesian approaches to estimation is presented and carefully discussed. The threshold problem - identifying the boundary between Bayes estimators which tend to outperform standard frequentist estimators and Bayes estimators which don’t - is formulated in an analytically tractable way. The formulation includes a specific (decision-theory based) criterion for comparing estimators. The centerpiece of the monograph, under quite general conditions, an explicit solution to the threshold is obtained for the problem of estimating a scalar parameter under squared error loss. The six chapters that follow address a variety of other contexts in which the threshold problem can be productively treated. Included are treatments of the Bayesian consensus problem, the threshold problem for estimation problems involving of multi-dimensional parameters and/or asymmetric loss, the estimation of nonidentifiable parameters, empirical Bayes methods for combining data from ‘similar’ experiments and linear Bayes methods for combining data from ‘related’ experiments. The final chapter provides an overview of the monograph’s highlights and a discussion of areas and problems in need of further research.Point Estimation from a Decision-Theoretic Viewpoint.
Tennis anyone? A glimpse at Game Theory.
Experimental data, decision rules and the risk function.
Point estimation as a decision problem; approaches to optimization.
An Overview of the Frequentist Approach to Estimation.
Preliminaries.
Minimum variance unbiased estimators.
Best linear unbiased estimators.
Best invariant estimators.
Some comments on estimation within restricted classes.
Estimators motivated by their behavior in large samples.
Robust estimators of a population parameter.
An Overview of the Bayesian Approach to Estimation.
Bayes’ Theorem.
The subjectivist view of probability.
The Bayesian paradigm for data analysis.
The Bayes risk.
The class of Bayes and “almost Bayes” rules.
The likelihood principle.
Conjugate prior distributions.
Bayesian robustness.
Bayesian asymptotics.
Bayesian computation.
Bayesian interval estimation.
The Threshold Problem.
Traditional approaches to comparing Bayes and frequentist estimators.
Logic.
Objectivity.
Asymptotics.
Ease of application.
Admissibility.
The treatment of high-dimensional parameters.
Shots across the bow.
Modeling the true state of nature.
A criterion for comparing estimators.
The threshold problem.
Comparing Bayesian and Frequentist Estimators of a Scalar Parameter.The word-length experiment.
A theoretical framework.
Empirical results.
Potpourri.
Discussion.
Conjugacy, Self-Consistency and Bayesian Consensus.
Another look at conjugacy.
Bayesian self-consistency.
An approach to the consensus problem.
Bayesian vs. Frequentist Shrinkage in Multivariate Normal Problems.
Preliminaries.
A solution to the threshold problem.
Discussion.
Comparing Bayesian and Frequentist Estimators under Asymmetric Loss.Estimating the mean of a normal distribution under Linex loss.
Estimating a linear combination of regression parameters.
Discussion.
The Treatment of Nonidentifiable Models.
The classical viewpoint.
The Bayesian treatment of nonidentifiability.
Estimation for a nonidentifiable binomial model.
On the efficacy of Bayesian updating in the binomial model.
On the efficacy of Bayesian updating in the nonparametric competing risks problem.
Bayesian estimation of a nonidentifiable parameter in a reliability context.
Improving on Standard Bayesian and Frequentist Estimators.
The empirical Bayes framework.
How to be a better Bayesian.
How to be a finer frequentist.
Combining Data from “Related” Experiments.A linear Bayesian approach to treating related experiments.
Modeling and linear Bayesian inference for data from related life testing experiments.
Discussion.
Fatherly Advice.
Where do I get off?
An overview.
Implications.
Desiderata.
Appendix: Standard Univariate Probability Models.