This graduate textbook covers topics in statistical theory essential for graduate students preparing for work on a Ph.D. degree in statistics. The first chapter provides a quick overview of concepts and results in measure-theoretic probability theory that are useful in statistics. The second chapter introduces some fundamental concepts in statistical decision theory and inference. Chapters 3-7 contain detailed studies on some important topics: unbiased estimation, parametric estimation, nonparametric estimation, hypothesis testing, and confidence sets. A large number of exercises in each chapter provide not only practice problems for students, but also many additional results. In addition to the classical results that are typically covered in a textbook of a similar level, this book introduces some topics in modern statistical theory that have been developed in recent years, such as Markov chain Monte Carlo, quasi-likelihoods, empirical likelihoods, statistical functionals, generalized estimation equations, the jackknife, and the bootstrap. Jun Shao is Professor of Statistics at the University of Wisconsin, Madison.
This book consists of four hundred exercises in mathematical statistics and their solutions, over 95% of which are in the author's Mathematical Statistics, Second Edition (Springer, 2003). For students preparing for work on a Ph.D. degree in statistics and instructors of mathematical statistics courses, this useful book provides solutions to train students for their research ability in mathematical statistics and presents many additional results and examples that complement any text in mathematical statistics. To develop problem-solving skills, two solutions and/or notes of brief discussions accompany a few exercises.
The exercises are grouped into seven chapters with titles matching those in the author's Mathematical Statistics. On the other hand, the book is stand-alone because exercises and solutions are comprehensible independently of their source, and notation and terminology are explained in the front of the book.
Readers are assumed to have a good knowledge in advanced calculus. A course in real analysis or measure theory is highly recommended. If this book is used with a statistics textbook that does not include probability theory, then knowledge in measure-theoretic probability theory is required.
Jun Shao is Professor of Statistics at the University of Wisconsin, Madison.
