A comparison of the Bayesian and frequentist approaches to estimation
The book under review is addressed to the audience of academics and practitioners who are open to using either frequentist or Bayesian methods in the considered problems, but would like to have a defensible basis for using one or the other. It will be appropriate as a text either for an advanced undergraduate course or for a graduate-level course or seminar. The book consists of 12 chapters and an appendix. The first three chapters present a review of decision-theoretic concepts and the basic tools and ideas of frequentist and Bayesian estimation. Then the author proceeds with the comparison of frequentist and Bayesian estimators in a variety of statistical contexts. The ``threshold problem'' is introduced in Chapter 4. It is the problem of identifying the boundary between the circumstances in which Bayes estimators tend to outperform frequentist ones and the complimentary circumstances in which the opposite is true. A criterion for comparing Bayes and frequentist estimators is then formulated. One-parameter problems estimated under squared error loss are considered in Chapter 5, and an explicit solution to the threshold problem, applicable to exponential families of sampling distributions and conjugate families of prior distributions, is given. In Chapter 6, the notion of conjugacy is further explored, both in the light of the concept of Bayesian self-consistency and as a tool for treating the Bayesian consensus problem. In Chapter 7, a multivariate version of the threshold problem is developed for comparing Bayesian and frequentist shrinkage. Generalizations of the threshold problem to estimation under an asymmetric loss function are considered in Chapter 8. In Chapter 9, the efficiency of Bayesian estimators of nonidentifiable parameters is examined through a concrete example in which a fully Bayesian version of the threshold problem is treated for a nonidentifiable binomial model. Nonparametric estimation in the context of competing risks and the estimation of the parameters of a nonidentifiable stress-strength model in reliability is also discussed. In Chapter 10, prescriptions are given for improving upon a Bayesian or frequentist estimator when data is available from one or more past similar experiments satisfying the empirical Bayes sampling assumptions. It is shown that such improvement is always possible. In Chapter 11, estimation problems are studied in which data is available from several related (rather than similar) experiments. It is shown that the strategy of employing information from past experiments is a possibility for improved estimation in the current experiment. Chapter 12 presents a summary and synthesis of the monograph's main topics and formulates a general set of recommendations and conclusions regarding the types of problems in which the Bayesian approach to estimation stands to provide reliable and preferred solutions. It concludes with comments on open problems of interest and promising directions for future research. An Appendix contains a list of standard univariate models with the parametrizations used in the book. The list of references consists of 147 items.
scientific article; zbMATH DE number 5768441