This page has been updated to reflect topics actually covered for future reference.

Course Details:

Instructor: Todd Kuffner

Lecture: 8:30-10am, Monday and Wednesday, Cupples I, Room 218

Required textbooks:

• (AL): Athreya & Lahiri's Measure Theory and Probability Theory, First Edition, Springer.
  
• (TPE): Lehmann & Casella's Theory of Point Estimation, Second Edition, Springer.
  
• (TSH): Lehmann & Romano's Testing Statistical Hypotheses, Third Edition, Springer.
  

We also made use the following references:

• Lehmann's Elements of Large Sample Theory
• Young & Smith's Essentials of Statistical Inference
• Petrov's Limit Theorems in Probability Theory
• Hall's The Bootstrap and Edgeworth Expansion
• DasGupta's Asymptotic Theory of Statistics and Probability
• Severini's Likelihood Methods in Statistics
• Shao's Mathematical Statistics
• Kleijn, van der Vaart and van Zanten's lecture notes on Bayesian Nonparametrics (related to forthcoming book by Ghosal & van der Vaart)
  
• David Hunter's lecture notes on asymptotic theory
  

Exams: 1 midterm and 1 final (also a Ph.D. qualifying exam for those students electing to take it)

Homework: there were homework assignments

In-class presentation: students were required to read a recent paper in a leading statistics journal and give a critical presentation of the methods, making reference to tools learned during Math 5061-5062. Two groups were formed (4 students each) and 40-minute presentations with slides were given on the following two papers:

• Marin, J.-M., Pillai, N.S., Robert, C.P. and J. Rousseau (2014). Relevant statistics for Bayesian model choice, Journal of the Royal Statistical Society B, 76 (5), 833-859.
  http://onlinelibrary.wiley.com/doi/10.1111/rssb.12056/abstract
• Martin, R. and C. Liu (2015). Conditional inferential models: combining information for prior-free probabilistic inference, Journal of the Royal Statistical Society B, 77 (1), 195-217.
  http://onlinelibrary.wiley.com/doi/10.1111/rssb.12070/abstract
  
  

Grades: 30% Homework, 25% Midterm, 35% Final, 10% In-class presentation

Topics List (reflecting what was actually covered):

1. Preliminaries: stochastic orders of magnitude, stochastic convergencem, characteristic functions, multivariate normal distribution, Kullback-Leibler divergence, pivots, asymptotically pivotal quantities
2. Optimal Estimation: asymptotic unbiasedness, consistency, various WLLN and SLLN
3. Convergence in Distribution: continuous mapping theorem, Helly-Bray theorem, dominated convergence theorem, Skorohod's representation theorem, the joys of Fatou's lemma
4. Asymptotic normality: Cramer-Wold theorem, Lindeberg and Lyapunov conditions, triangular arrays, Lindeberg-Feller CLT
5. Refinements: Berry-Esseen theorem, Edgeworth expansions
6. Toolbox: delta method, variance stabilizing transformations
7. Applications: asymptotic distributions of sample moments and order statistics
8. Likelihood theory: MLE consistency, Fisher information, asymptotic normality, MLE with many parameters; profile likelihood; Wald, score and LR tests; Wilks' theorem
9. Bayesian Parametric Asymptotics: Scheffe's theorem, Bernstein-von Mises theorem, Doob's theorem, difference between Bayes estimates and MLE
10. Optimal Hypothesis Testing: consistency for tests, consistency under local alternatives, asymptotic relative efficiency (in the context of testing)
11. Frequentist Nonparametric Methods: consistency of nonparametric bootstrap;  why and how the nonparametric bootstrap can be used to achieve bias correction; why and how the nonparametric bootstrap can be used to achieve refinement of first-order asymptotic distribution theory for asymptotically standard normal pivots
12. Bayesian Nonparametric Methods: key concepts: prior construction (random measures), Dirichlet processes, consistency (Schwartz's theorem), posterior rates of contraction