University: Massachusetts Institute of Technology
Instructors: Prof. Richard Dudley
Course Number: 18.466
This graduate level mathematics course covers decision theory, estimation, confidence intervals, and hypothesis testing. The course also introduces students to large sample theory. Other topics covered include asymptotic efficiency of estimates, exponential families, and sequential analysis.
One semester beginning graduate real analysis and measure theory, as in MIT course 18.125, specifically Chapters 1-5 of the book Real Analysis and Probability, by R. Dudley, 2nd ed., Cambridge University Press, 2002. One or more previous courses in probability or statistics will be helpful background but are not required.
Chapter 1: Decision Theory and Testing Simple Hypotheses
Chapter 2: Sufficiency and Estimation
Chapter 3: Bayes, Maximum Likelihood and M-estimation
The term paper should be an expository paper of about 10 printed pages on some material related to the course which has either not been written up previously in books but only in one or more papers, or perhaps in some specialized book(s), but it should not be a "standard topic" found in multiple statistics texts. The level should be such that the other students in the class could understand the paper about as well as the printed lecture notes.
A first draft of the paper is due in session #37. It will be returned with comments in session #38. The final version will be due in session #39. Please try to meet the schedule if at all possible, which probably means choosing a topic rather soon so you can get started on the paper.
In the December 2000 issue of the Journal of the American Statistical Association ("JASA"), on pp. 1269-1368, are a set of vignettes on various topics in statistics. Here you may find information about possible paper topics.
I don’t suggest that any vignette by itself would be a good basis for a paper, but it would give you an entry point into the literature of a topic. It might indicate what books already exist and on what subtopics there are interesting papers whose material has not been written up yet into books. You might then pick one or two papers as bases for yours. It could be good to scan a few papers before making your decision. When you’ve chosen a topic, please let me know. I would like to have no more than one student writing on a given specific topic, although several of the topics are broad enough to have more than one suitable subtopic.
The topics of the vignettes are:
2. Statistical Decision Theory
3. Markov Chain Monte Carlo
4. Empirical Bayes
5. Linear and Log-linear Models
6. The Bootstrap
7. Nonparametric Modeling
8. Gibbs Sampling
9. The Variable Selection Problem
10. Robust Nonparametric Methods
11. Hierarchical Models
12. Hypothesis Testing
13. Generalized Linear Models
14. Missing Data
17. Conditioning, Likelihood and Coherence
18. Time Series
19. Information Theoretic Approaches
20. Measurement Error Models
21. Higher-order Asymptotic Approximation
Of these, numbers 7, 10 and 15 seem to relate more to 18.458 (a course on nonparametrics and robustness) than to 18.466. Numbers 5, 9, 11, 13, 18, and 20 are topics that we haven’t approached apparently in the course, which doesn’t necessarily mean they would be unsuitable paper topics.
Possible topics not on the vignette list are:
1. Contiguity (work of Le Cam and others)
2. The Likelihood Principle and Related Matters (cf. lecture note volume by Berger and Wolpert)
3. Model Selection
4. Ancillarity (work by O. Barndorff-Nielsen and others)