Wei Lin @ PKU

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