Syllabus
18.655 - Mathematical Statistics
We will cover the following topics:
Statistical decision theory
Exponential families
Data reduction: sufficiency, ancillarity, completeness
Bayes and minimax estimation
Hypothesis testing and confidence intervals
Maximum likelihood estimation
Large sample theory
Resampling methods: bootstrap and permutation tests
Sequential testing
A tentative schedule is as follows.
Reading assignments are from Keener, unless otherwise specified.
W - High Dimensional Statistics by Wainwright (see references)
| Date | Topic | Reading |
| 2/6 | Introduction, probability models, risk functions | 1, 3.1 |
| 2/11 | Exponential families | 2 |
| 2/13 | Sufficient statistics | 2, 3.2 |
| 2/19 (Monday schedule) | Minimal sufficiency, completeness | 3.4, 3.5, 3.6 |
| 2/20 | Rao-Blackwell theorem, UMVU estimation | 3.6, 4.1, 4.2 |
| 2/25 | UMVU estimation, Cramer-Rao | 4.2, 4.2, 4.5 |
| 2/27 | Information and estimation | 4.5, 4.6 |
| 3/4 | Bayesian estimation | 7.1, 7.2 |
| 3/6 | No class | - |
| 3/11 | Hypothesis testing, Neyman-Pearson lemma | 12.1, 12.2, 12.3, 12.4 |
| 3/13 | UMP tests | 12.3, 12.4, 12.5, 12.7 |
| 3/18 | UMP unbiased tests | 13.1, 13.2, 13.3 |
| 3/20 | Midterm (in class) | - |
| 3/25 | Spring break | - |
| 3/27 | Spring break | - |
| 4/1 | Asymptotic concepts | 8.1, 8.2, 8.3, |
| 4/3 | Maximum likelihood | 8.3, 8.4 |
| 4/8 | Relative efficiency | 8.5, 9.1, 9.2 Stigler, Efron |
| 4/10 | Consistency of MLE | 9.1, 9.2, 9.3 |
| 4/15 | Holiday - Patriot's day | - |
| 4/17 | Asymptotic normality of MLE | 9.1, 9.2, 9.3 |
| 4/22 | Asymptotic tests and CIs | 9.4, 9.5, 9.7 |
| 4/24 | Minimax lower bounds: Le Cam | W15.1 |
| 4/29 | Minimax lower bounds: proof of Le Cam | W15.2 |
| 5/1 | Minimax lower bounds using Fano | W15.3 |
| 5/6 | Proof of Fano's method | W15.3.2, papers/baraudFano.pdf |
| 5/8 | Proof of Fano's method using information theory | W15.3 |
| 5/13 | Review | |
| 5/15 | What we did not cover |
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