Math 12640: Advanced Theory of Statistics
Course Description
This is a core course in statistical theory for beginning graduate students in statistics, probability, and related fields where a sound
understanding of statistical principles is essential.
Syllabus
Lectures and Assignments
Week | Date | Topics | References | Assignments | Notes |
1 | 9/11 | Statisitical models | Shao 2.1.1, 2.1.2 | | |
| 9/13 | Exponential families | Shao 2.1.3 | 2.9, 2.11, 2.14 | |
2 | 9/18 | Statistial decision theory | Shao 2.3 | 2.63, 2.74, 2.80, 2.84, 2.95 | Homework 1 due 9/25 |
3 | 9/25 | Sufficient statistics | Shao 2.2.2 | 2.33 | |
| 9/27 | Complete statistics | Shao 2.2.3 | 2.45, 2.47, 2.52, 2.60 | |
4 | 10/2 | National Day | | | |
5 | 10/9 | Criteria of estimation | Shao 2.4.1, 2.5.1, 2.5.2 | 2.103, 2.117 | |
| 10/10 | Unbiased estimation | Shao 3.1.1, 3.1.2 | 3.2, 3.5, 3.26 | Homework 2 due 10/16 |
6 | 10/16 | Information inequality; Method of moments | Shao 3.1.3, 3.1.4, 3.5.2 | 3.38, 3.40, 3.109 | |
7 | 10/23 | Bayes actions and estimators | Shao 4.1.1–4.1.3 | 4.7, 4.12, 4.34, 4.35 | |
| 10/25 | Minimax estimators | Shao 4.3.1 | 4.69, 4.77 | Homework 3 due 10/30 |
8 | 10/30 | Discussion | | | |
9 | 11/6 | Midterm exam | | | Mean = 65, median = 64, Q1 = 52, Q3 = 81, high score = 98 |
| 11/8 | Maximum likelihood | Shao 4.4.1 Bickel & Doksum 2.3 | 4.100, 4.104, 4.110 | |
10 | 11/13 | MLEs in generalized linear models; Superefficiency | Shao 4.4.2, 4.5.1 | 4.117 | |
11 | 11/20 | Asymptotic efficiency of MLEs; One-step MLEs | Shao 4.5.2, 4.5.3 | 4.125, 4.134 | |
| 11/22 | Hypothesis testing; The Neyman–Pearson lemma | Shao 6.1.1 | 6.3, 6.6 | Homework 4 due 11/27 |
12 | 11/27 | Uniformly most powerful tests; UMP unbiased tests | Shao 6.1.2–6.2.2 | 6.16, 6.18, 6.27, 6.41 | |
13 | 12/4 | Likelihood ratio tests; Bayes tests | Shao 6.4.1, 6.4.4 | 6.86, 6.106 | |
| 12/6 | Asymptotic tests; 𝜒2-tests | Shao 6.4.2, 6.4.3 | 6.96, 6.100 | Homework 5 due 12/20 |
14 | 12/11 | Construction of confidence sets | Shao 7.1.1., 7.1.2 | 7.2, 7.25 | |
15 | 12/18 | Optimality of confidence sets; Bayesian credible sets | Shao 7.2.1, 7.2.2, 7.1.3 | 7.28, 7.32, 7.44 | |
| 12/20 | Asymptotic confidence sets; Prediction sets; Simultaneous confidence intervals | Shao 7.3.1, 7.3.2, 7.1.4, 7.5.1 | 7.30, 7.66, 7.102 | Homework 6 due 12/25 |
16 | 12/25 | Discussion | | | |
18 | 1/8 | Final exam | | | Mean = 59, median = 58, Q1 = 52, Q3 = 64, high score = 77
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