This course continues and extends the study of statistical theory initiated in Advanced Theory of Statistics. After introducing modern
mathematical tools including empirical processes, concentration inequalities, and random matrices, we will discuss general theory and methodology
for M- and Z-estimators, semiparametric and nonparametric models, and high-dimensional models, from both asymptotic and nonasymptotic points
of view. Working through this course will help students develop a comprehensive understanding of modern statistical theory and acquire the
necessary knowledge and skills for conducting research in statistics and data science.
| Week | Date | Topics | References | Assignments and Notes |
| 1 | 3/10 | Overview, statistical motivations | Wainwright Chap. 1, Sec. 4.1 | |
| 3/12 | Stochastic convergence in metric spaces, classical empirical processes | vdV Chap. 18, Sec. 19.1 | |
| 2 | 3/17 | Empirical processes via bracketing and uniform entropy | vdV Sec. 19.2, Kosorok Secs. 8.2–8.4 | |
| 3 | 3/24 | Preservation results, random functions | Kosorok Sec. 8.2, Chap. 9, vdV Sec. 19.4 | |
| 3/26 | Functional delta method, classical concentration inequalities | vdV Chap. 20, Wainwright Sec. 2.1 | Homework 1 |
| 4 | 3/31 | Martingale concentration inequalities | Wainwright Sec. 2.2 | |
| 5 | 4/7 | Optimal transport, concentration for empirical processes | Wainwright Secs. 3.3, 3.4 | |
| 4/9 | Wishart matrices and sub-Gaussian ensembles | Wainwright Secs. 6.1–6.3 | |
| 6 | 4/14 | Concentration for general random matrices, M- and Z-estimators | Wainwright Sec. 6.4, vdV Sec. 5.1 | Homework 2 |
| 7 | 4/21 | Consistency and asymptotic normality | vdV Secs. 5.2, 5.3 | |
| 4/23 | Maximum likelihood estimators | vdV Sec. 5.5 | |
| 8 | 4/28 | Semiparametric models | vdV Secs. 25.1, 25.2 | |
| 9 | 5/5 | Midterm exam | | Mean = 50, median = 50, Q1 = 41, Q3 = 60, high score = 68 |
| 5/7 | Efficiency and information | vdV Secs. 25.3, 25.4 | |
| 10 | 5/12 | Semiparametric inference via efficient score equations | vdV Sec. 25.8 | |
| 11 | 5/19 | Semiparametric inference via estimating equations and maximum likelihood | vdV Secs. 25.9, 25.10 | Homework 3 |
| 5/21 | Nonparametric least squares, prediction bounds | Wainwright Secs. 13.1, 13.2 | |
| 12 | 5/26 | Oracle inequalities | Wainwright Sec. 13.3 | |
| 13 | 6/2 | Regularized M-estimators, decomposibility | Wainwright Secs. 9.1, 9.2 | |
| 6/4 | Restricted strong convexity | Wainwright Secs. 9.3, 9.4 | |
| 14 | 6/9 | Sparse regression, matrix estimation | Wainwright Sec. 9.5, Chap. 10 | Homework 4 |
| 15 | 6/16 | Mimimax lower bounds, Le Cam's method | Wainwright Secs. 15.1, 15.2 | |
| 6/18 | Fano's method | Wainwright Sec. 15.3 |
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