Mathematical Statistics
This book is intended for a course entitled Mathematical Statistics o?ered at the Department of Statistics, University of Wisconsin-Madison. This course, taught in a mathematically rigorous fashion, covers essential - terials in statistical theory that a
- PDF / 4,389,105 Bytes
- 543 Pages / 325.984 x 411.914 pts Page_size
- 50 Downloads / 178 Views
Preface This book is intended for a course entitled Mathematical Statistics offered at the Department of Statistics, University of Wisconsin-Madison. This course, taught in a mathematically rigorous fashion, covers essential materials in statistical theory that a first or second year graduate student typically needs to learn as preparation for work on a Ph.D. degree in statistics. The course is designed for two 15-week semesters, with three lecture hours and two discussion hours in each week. Students in this course are assumed to have a good knowledge of advanced calculus. A course in real analysis or measure theory prior to this course is often recommended. Chapter 1 provides a quick overview of important concepts and results in measure-theoretic probability theory that are used as tools in the rest of the book. Chapter 2 introduces some fundamental concepts in statistics, including statistical models, the principle of sufficiency in data reduction, and two statistical approaches adopted throughout the book: statistical decision theory and statistical inference. Each of Chapters 3 through 7 provides a detailed study of an important topic in statistical decision theory and inference; Chapter 3 introduces the theory of unbiased estimation; Chapter 4 studies theory and methods in point estimation under parametric models; Chapter 5 covers point estimation in nonparametric settings; Chapter 6 focuses on hypothesis testing; and Chapter 7 discusses interval estimation and confidence sets. The classical frequentist approach is adopted in this book, although the Bayesian approach is also introduced (§2.3.2, §4.1, §6.4.4, and §7.1.3). Asymptotic (large sample) theory, a crucial part of statistical inference, is studied throughout the book, rather than in a separate chapter. About 85% of the book covers classical results in statistical theory that are typically found in textbooks of a similar level. These materials are in the Statistics Department’s Ph.D. qualifying examination syllabus. This part of the book is influenced by several standard textbooks, such as Casella and Berger (1990), Ferguson (1967), Lehmann (1983, 1986), and Rohatgi (1976). The other 15% of the book covers some topics in modern statistical theory
vii
viii
Preface
that have been developed in recent years, including robustness of the least squares estimators, Markov chain Monte Carlo, generalized linear models, quasi-likelihoods, empirical likelihoods, statistical functionals, generalized estimation equations, the jackknife, and the bootstrap. In addition to the presentation of fruitful ideas and results, this book emphasizes the use of important tools in establishing theoretical results. Thus, most proofs of theorems, propositions, and lemmas are provided or left as exercises. Some proofs of theorems are omitted (especially in Chapter 1), because the proofs are lengthy or beyond the scope of the book (references are always provided). Each chapter contains a number of examples. Part of them are designed as materials covered in the discussion section of t
Data Loading...
