Book Reviews

Abstract

By the authors' admission, this monograph is not intended as 'a complete up-to-date account of the Bayesian literature in finite population sampling', but is more a reflection of the authors' 'personal interests'. By inspection of the authors' publications, this is obvious: the focus of the book is on Bayesian sampling with different degrees of prior information (without specifying a prior distribution and using a pseudo-posterior) and resultant admissibility issues.I was surprised. Given such a narrow focus, I feared an esoteric and disjointed treatment of this topic. Instead, I was treated to well-written discussion of the links between standard frequentist and the new Bayesian results, easily-followed graduation between 'Bayesian foundations', the noninformative approach and hierarchical Bayes models, and comprehensive motivation for the more technical ideas in the book. As welcome reinforcement, I was constantly told what to expect from a section and then treated to a discussion of what I had been told. Theorems and proofs were reasonably well integrated into the text, although obviously this became a little more difficult as the complexity of the material increased. Although examples were not a strong feature of the book, simple examples illustrated early points well, and later chapters briefly described 'real' applications. Computational issues, including simulation and MCMC, were also addressed.The basis of the book is the 'Polya posterior', a predictive distribution for the unobserved units in the population given the observed units. If the unobserved and observed units are considered to be exchangeable, this has correspondence with the Beta and Dirichlet distributions and allows a noninformative stepwise Bayesian justification of the usual frequentist inferential procedures. Extensions to the incorporation of different kinds of prior information (about the entire population or about each member of the population, or auxiliary information), stratification, nonresponse, nonparametric models and nonexchangeability are also considered. The reader then progresses to applications of empirical and hierarchical Bayes methods, which the authors particularly promote for small area estimation. Connection is made between the now familiar stepwise Bayes estimators and empirical Bayes estimators, and the hierarchical model is used to connect model-and designbased estimators. Multistage sampling, nested models and generalized linear models are variously discussed.In my opinion, the book is well worth reading and is quite readable, particularly as a reference for a statistician who is researching in the area of finite population sampling, as extension material for an able student, or as a base from which a practitioner might develop new insights and ideas about sampling.