Statistical Decision Theory
Estimation, Testing, and Selection
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For advanced graduate students, this book is a one-stop shop that presents the main ideas of decision theory in an organized, balanced, and mathematically rigorous manner, while observing statistical relevance. All of the major topics are introduced at an elementary level, then developed incrementally to higher levels. The book is self-contained as it provides full proofs, worked-out examples, and problems. 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. With its broad coverage of decision theory, this book fills the gap between standard graduate texts in mathematical statistics and advanced monographs on modern asymptotic theory.
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.
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