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Mathematical Statistics - Jun Shao 高清英文版 非扫描
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 mathematical
statistics. 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
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. Some of them are designed as materials covered in the discussion
section of this course, which is typically taught by a teaching assistant (a
senior graduate student). The exercises in each chapter form an important
part of the book. They provide not only practice problems for students,
but also many additional results as complementary materials to the main
text.
The book is essentially based on (1) my class notes taken in 1983-84
when I was a student in this course, (2) the notes I used when I was a
teaching assistant for this course in 1984-85, and (3) the lecture notes I
prepared during 1997-98 as the instructor of this course. I would like to
express my thanks to Dennis Cox, who taught this course when I was
a student and a teaching assistant, and undoubtedly has influenced my
teaching style and textbook for this course. I am also very grateful to
students in my class who provided helpful comments; to Mr. Yonghee Lee,
who helped me to prepare all the figures in this book; to the Springer-Verlag
production and copy editors, who helped to improve the presentation; and
to my family members, who provided support during the writing of this
book.
Madison, Wisconsin Jun Shao
January 1999
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