Statsitical Decision Theory: Estimation, Testing, Selection

This monograph is written for advanced Master’s students, Ph.D. students, and researchers in mathematical statistics and decision theory. It should be useful not only as a basis for graduate courses, seminars, Ph.D. programs, and self-studies, but also as a reference tool. Attheveryleast,readersshouldbefamiliar withbasicconceptscoveredin both advanced undergraduate courses on probability and statistics and int- ductory graduate-level courses on probability theory, mathematical statistics, and analysis. Most statements and proofs appear in a form where standard arguments from measure theory and analysis are su?cient. When additional information is necessary, technical tools, additional measure-theoretic facts, and advanced probabilistic results are presented in condensed form in an - pendix. In particular, topics from measure theory and from the theory of weak convergence of distributions are treated in detail with reference to m- ern books on probability theory, such as Billingsley (1968), Kallenberg (1997, 2002), and Dudley (2002). Building on foundational knowledge, this book acquaints readers with the concepts of classical ?nite sample size decision theory and modern asymptotic decision theory in the sense of LeCam. To this end, systematic applications to the ?elds of parameter estimation, testing hypotheses, and selection of po- lations are included. Some of the problems contain additional information in order to round o? the results, whereas other problems, equipped with so- tions, have a more technical character. The latter play the role of auxiliary results and as such they allow readers to become familiar with the advanced techniques of mathematical statistics.

This monograph is written for advanced graduate students, Ph.D. students, and researchers in mathematical statistics and decision theory. All major topics are introduced on a fairly elementary level and then developed gradually to higher levels. The book is self-contained as it provides full proofs, worked-out examples, and problems. It can be used as a basis for graduate courses, seminars, Ph.D. programs, self-studies, and as a reference book.

The authors present a rigorous account of the concepts and a broad treatment of the major results of classical finite sample size decision theory and modern asymptotic decision theory. Highlights are systematic applications to the fields of parameter estimation, testing hypotheses, and selection of populations. With its broad coverage of decision theory that includes results from other more specialized books as well as new material, this book is one of a kind and fills the gap between standard graduate texts in mathematical statistics and advanced monographs on modern asymptotic theory.

One goal is to present a bridge from the classical results of mathematical statistics and decision theory to the modern asymptotic decision theory founded by LeCam. The striking clearness and powerful applicability of LeCam’s theory is demonstrated with its applications to estimation, testing, and selection on an intermediate level that is accessible to graduate students. Another goal is to present a broad coverage of both the frequentist and the Bayes approach in decision theory. Relations between the Bayes and minimax concepts are studied, and fundamental asymptotic results of modern Bayes statistical theory are included. The third goal is to present, for the first time in a book, a well-rounded theory of optimal selections for parametric families.

Friedrich Liese, University of Rostock, and Klaus-J. Miescke, University of Illinois at Chicago, are professors of mathematical statistics who have published numerousresearch papers in mathematical statistics and decision theory over the past three decades.