Math 12640: Advanced Theory of Statistics
Course Description
This is a core course in statistical theory for beginning graduate students in statistics, probability, and application fields where a sound
understanding of statistical principles is essential.
Syllabus
Lectures and Assignments
| Week | Date | Topics | References | Assignments | Notes |
| 1 | 9/12 | Statisitical models | 1.1 | 1.1.5, 1.1.9 | For a formal discussion on ‘‘What is a statistical model?’’ using category theory, see McCullagh (2002). For a highly nontrivial example of identifiability issues invovling algebraic geometry, see Chandrasekaran et al. (2012). |
| 9/14 | Bayesian models; Decision theory | 1.2, 1.3 | 1.2.14, 1.2.15, 1.3.3, 1.3.11, 1.3.17 | An alternative inferential framework to the frequentist and Bayesian paradigms was presented by Martin & Liu (2013). |
| 2 | 9/19 | Prediction; Sufficiency | 1.4, 1.5 | 1.4.11, 1.5.5, 1.5.7 | |
| 3 | 9/26 | Minimal sufficiency; Completeness | Shao 2.2.2, 2.2.3 | 1.5.10, Shao 2.45, 2.47, 2.52, 2.60 | Minimal sufficient statistics exist under weak assumptions; see Lehmann & Casella, p. 37. |
| 9/28 | Exponential families | 1.6 | 1.6.4, 1.6.18 | Curved exponential families can be very useful for modeling dependent data such as networks; see, e.g., Hunter & Handcock (2006). |
| 4 | 10/3 | National Day | | | |
| 5 | 10/10 | M- and Z-estimators; Method of moments | 2.1.1 | 2.1.1, 2.1.9 | |
| 10/12 | Plug-in and extension principles; Maximum likelihood | 2.1.2, 2.2 | 2.2.12, 2.2.15, 2.2.19, 2.2.35 | The epic story of maximum likelihood was told by Stigler (2007). |
| 6 | 10/17 | MLEs in exponential families; Coordinate descent | 2.3, 2.4.2 | 2.3.7, 2.3.10, 2.4.3 | A recent review of coordinate descent algorithms can be found in Wright (2015). |
| 7 | 10/24 | Bayes procedures | 3.2, Shao 4.1.3 | 3.2.5, 3.2.8 | |
| 10/26 | Minimax procedures | 3.3, Shao 4.3.1 | 3.3.5, 3.3.9, 3.3.12 | |
| 8 | 10/31 | Unbiased estimation; Information inequality | 3.4, Shao 3.1.1–3.1.3 | 3.4.2, 3.4.12, Shao 3.2, 3.5 | |
| 9 | 11/7 | Midterm exam | | | Mean = 66, median = 69, Q1 = 57, Q3 = 77, high score = 98 |
| 11/9 | Hypothesis testing; The Neyman–Pearson lemma | 4.1, 4.2, Shao 6.1.1 | 4.1.6, 4.2.3, 4.2.7 | The origins of the 0.05 level of significance were explained by Cowles & Davis (1982). |
| 10 | 11/14 | Uniformly most powerful tests | 4.3, Shao 6.1.2, 6.1.3 | 4.3.4, 4.3.9 | |
| 11 | 11/21 | Likelihood ratio tests | 4.9, Shao 6.4.1 | 4.9.4, 4.9.9 | Perlman & Wu (1999) defended likelihood ratio tests against unbiased and more powerful size-𝛼 tests. |
| 11/23 | Bayes tests; Confidence sets via the method of pivots | 4.4, Shao 6.4.4, 7.1.1 | 4.4.2, Shao 6.106 | |
| 12 | 11/28 | Confidence sets via inverting tests; Lengths of CIs | 4.5, Shao 7.1.2, 7.2.1 | 4.4.6, 4.5.5, Shao 7.35 | |
| 13 | 12/5 | Uniformly most accurate confidence sets; Bayesian credible sets; Prediction sets; Simultaneous CIs | 4.4, 4.6–4.8, Shao 7.1.3, 7.1.4, 7.2.2, 7.5.1 | 4.4.15, 4.6.1, 4.7.2, 4.8.3 | |
| 12/7 | Stochastic convergence | Shao 1.5.1–1.5.3 | Shao 1.127, 1.138 | |
| 14 | 12/12 | Asymptotic theory; Consistency | 5.1–5.3, Shao 1.5.4–1.5.6, 2.5.1 | 5.2.4, 5.3.25 | |
| 15 | 12/19 | Asymptotic criteria and inference | Shao 2.5.2, 2.5.3 | Shao 2.117, 2.124 | |
| 12/21 | Superefficiency; Asymptotic efficiency of MLEs | 5.4, 6.2, Shao 4.5 | 5.4.4, Shao 4.111, 4.125 | Vovk (2009) approached the phenomenon of superefficiency from an algorithmic perspective and showed that sets of points of superefficiency are countable. |
| 16 | 12/26 | Asymptotic tests; 𝜒2-tests | 6.3, 6.4, Shao 6.4.2, 6.4.3 | Shao 6.96, 6.100 | |
| 17 | 1/2 | Final exam: 8:30–10:30 am, 404 Classroom Building 2 | | | Mean = 57, median = 59, Q1 = 48, Q3 = 65, high score = 82
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