Designing Economic Mechanisms

methodology for obtaining regular asymptotic linear AIPWCC (augmented inverse probability weighted complete case) estimators. Additionally, Chapter 9 includes a presentation of the relationship between monotone coarsening and censoring. Chapters 10 and 11 develop methodology for obtaining efficient and robust estimators. Initially optimal influence functions, whose structure yields to the space of double robust influence functions, are identified. Then the discussion is divided into three parts, according to the structure of coarsening (namely two levels of missingness), monotone coarsening, and nonmonotone coarsening. The concepts are promoted nicely for every different case and the corresponding estimators are developed via both theory and intuition. Chapter 11 considers efficient estimation in the class of double robust estimators. The ideas are based on AIPWCC estimators, but as the author shows there are computational problems for their implementation. Motivated by the preceding results, Chapter 12 develops estimation within a restricted class of AIPWCC estimators. In particular, detailed proof of the form of the estimating equations is provided in two cases. Initially, it is assumed that the estimating function belongs to the q-replicating linear subspace of the space of influence functions and the q-replicating linear subspace of the augmentation space, where both of the spaces are assumed to be linear and finite dimensional. The second case considers the q-replicating linear subspace of the space of influence functions to be finite dimensional, while there is no restriction on the augmentation space. Examples are worked out in great detail for both cases. Chapter 13 demonstrates the theory to the problem of estimating the average causal treatment effect. The idea is based on the so-called stable unit treatment value assumption, which implies that estimation of the average causal treatment effect is equivalent to estimation with missing data. Estimators are derived in great detail, reinforcing in this way the previous results. The last chapter examines the asymptotic properties of multiple imputation estimators. The prerequisites vary in level and depth as the material advances, but a graduate class in statistics at the level of Casella and Berger (1990) suffices to follow the exposition. The writing style is excellent, and all the main concepts and ideas are presented in a clear and pedagogical way. For instance, a detailed study of both restricted moment and logistic models throughout the book is instrumental in illustrating the abstract theory to some standard data analysis tools. At the end of almost each chapter, there are problems and a summary, which recapitulates notation usage and important results. The author should have put more emphasis on real data examples—applications are missing from the presentation. This book is suitable for an advanced graduate course or for self-study by doctoral students or researchers in statistics and biostatistics. It provides a valuable resource because it contains an up-to-date literature review and an exceptional account of state of the art research on the necessary theory. Overall, Semiparametric Theory and Missing Data is an excellent addition to the literature and, without any hesitation, I recommend it to any professional statistician.