The following information is for Spring 2002:

General Description
This course is a detailed introduction to the theory of statistical inference. Topics include maximum likelihood and various other methods for estimation and hypothesis testing, especially for linear models. Most assignments involve solving problems and/or deriving elementary propositions.

Prerequisites
Math 402 makes extensive use of the methods of mathematical probability. Students should not take Math 402 until they have completed Math 401.

Mathematical statistics is concerned with the theoretical principles and methodologies that academic statisticians use to derive procedures for analyzing data. Thus, Math 402 is directly concerned with the theory of statistics and only indirectly concerned with data analysis. Students who desire a single, self-contained semester of applied statistics should take Math 308 instead of Math 402.

Math 402 is intended for mathematically inclined students who would like to understand the rationales that underlie many statistical methods. It can also serve to introduce math concentrators to an important mathematical science. Math 308 is not a prerequisite for Math 402; however, students who take Math 308 before Math 402 will have a better sense of why the theory developed in Math 402 is important.

Basic Information
Math 402 will meet on Tuesdays and Thursdays, from 9:30 to 10:50 a.m., in Room 306 of Jones Hall. The final exam is scheduled for 8:30 a.m. on Tuesday, May 7.

Tentative Office Hours
Tuesdays, 2:00 to 3:30 p.m., Wednesdays, 10:00 to 11:30 a.m., or by appointment, in Jones 127.

Attendance
Class attendance is not formally required, but it is strongly encouraged. Ignorance of supplementary material presented---or announcements made---by the instructor due to absence from class is never excusable. In class you are expected to behave appropriately, e.g. please refrain from conversing with other students while the instructor is lecturing.

Text
INTRODUCTION TO STATISTICAL THEORY, 1971, by P.G. Hoel, S.C. Port, and C.J. Stone. This book is available at the William & Mary bookstore. It will serve as the primary text, but I will supplement it with materials from a variety of other sources.

Syllabus
I plan to cover the following topics:

• Lecture 1. Overview of Statistical Inference. The basic ideas that underlie point estimation, set estimation, and hypothesis testing.
• Lecture 2. Inner Products and Norms. Definitions, Cauchy-Schwartz inequality, angles, examples.
• Lecture 3. Vector Calculus. Derivatives, Taylor polynomials, Newton's method.
• Lecture 4. Optimization. First- and second-order conditions for unconstrained minimizers, Newton's method, convexity, first-order conditions for constrained minimizers, projection into closed convex sets.
• Lecture 5. Linear Least Squares. The normal equations and some caveats about using them.
• Lecture 6. Statistical Models. Regularity, exponential families, sufficiency.
• Lecture 7. Statistical Models. Location, scale, and location-scale families; strong unimodality.
• Lecture 8. The Plug-In Principle. Statistical functionals, empirical distributions, bootstrapping.
• Lecture 9. Decision-Theoretic Criteria. Unbiased, equivariant, minimax, and Bayes estimation.
• Lecture 10. The Method of Moments and the Method of Maximum Likelihood.
• Lecture 11. Computing Maximum Likelihood Estimates.
• Lecture 12. Confidence Sets. (Section 2.7)
• Lecture 13. Hypothesis Testing. (Chapter 3)
• Lecture 14. The Neyman-Pearson Lemma. (Section 3.1)
• Lecture 15. Uniformly Most Powerful Tests. (Section 3.2)
• Lecture 16. Likelihood Ratio Tests. (Section 3.4)
• Lecture 17. Student's t-Tests. (Section 3.4.1)
• Lecture 18. Testing Goodness-of-Fit. (Section 3.5)
• Lecture 19. Testing Distributional Assumptions. (Section 3.5)
• Lecture 20. Testing Independence in a Contingency Table. (Section 3.5)
• Lecture 21. Simple Linear Regression. (Sections 2.9 and 4.1)
• Lecture 22. Linear Models. Matrix formulations of simple linear regression, multiple linear regression, 1-sample problems, 2-sample problems, k-sample problems. (Section 4.4)
• Lecture 23. Least Squares Estimation. The Gauss-Markov Theorem. (Section 4.5)
• Lecture 24. The General Linear Hypothesis. Formulation and application to various linear models. (Section 5.1)
• Lecture 25. Testing the General Linear Hypothesis. (Sections 5.1.1 and 5.1.2)
• Lecture 26. Inferences for Regression Coefficients. (Section 5.2)
• Lecture 27. Using Linear Models: A Case Study.