Class meetings, Spring 2018: Tuesday and Thursday 9:30am – 10:45am in Hanes 130
Prerequisites: STOR 654 (Theoretical Statistics I) and STOR 634 (Introduction to Measure Theoretic Probability), basic real analysis, and linear algebra.
Registration: Enrollment and registration for the course is handled online.
Instructor: Andrew B. Nobel
Office: Hanes 308 Email: email@example.com Phone: 919-962-1352.
Office Hours: TBA
TA: Leo Liu
Office: Hanes B-50 Email: firstname.lastname@example.org
Audience and Goals: STOR 655 is the second course in the first year graduate theoretical statistics sequence. The course is targeted to PhD students in the Statistics and Operations Research (STOR) department, but may be appropriate for MS students in STOR, and for students in other departments with appropriate mathematical and statistical backgrounds.
The goal of the course is to introduce students to some of the key results in theoretical statistics, and to the mathematical techniques that underly them. Key results include the consistency and asymptotic normality of method of moments and maximum likelihood estimation, the limiting distribution of likelihood ratio tests, Gaussian extreme value theory, and concentration inequalities for bounded and Gaussian random variables. Key techniques include weak convergence, the delta method, Taylor series, use of Jensen and related inequalities, Gaussian integration by parts, symmetrization, and contraction.
The first part of the course will be devoted to classical large sample theory. The second part of the course will be devoted to a selection of material that is more closely aligned with modern, high-dimensional inference procedures. Most topics will be self-contained, results being derived from first principles and the prerequisite material.
Text: The primary text for the first part of the class is “A Course in Large Sample Theory” by T.S. Ferguson. Material for the second part will come from course notes and online sources.
Homework policy: Homework problems will be assigned regularly throughout the semester, usually every week. Each homework assignment will be graded: late/missed homeworks will receive a grade of zero. Students are welcome to discuss the homework problems with other members of the class, but should prepare their final answers on their own. If you have any questions concerning the grading of homework, please speak first with the TA. If you are absent from class when an assignment is returned, you can get your homework from the TA during their office hours.
Attendance: Students are expected to attend all lectures. If you are unable to attend a lecture, please let the instructor know and make plans to get the notes from another student in the class.
Grading: Grading will be based on homeworks, an in-class midterm, and an in-class final exam, using the weights below.
Note: The final exam will be given at the time and date specified by the UNC Final Exam Schedule.
“Statistical Inference” by G. Casella and R. Berger. Provides a good background on probability and inference.
“Asymptotic Statistics” by A. van der Vaart. A more advanced treatment of the material in the class.
“Multivariate Analysis”, by K.V. Mardia, J.T. Kent and J.M. Bibby.
“Mathematical Statistics”, Second Edition, by P.J. Bickel and K.A. Doksum, Prentice Hall, 2001.
Honor Code: Students are expected to adhere to the UNC honor code at all times.
1. Large Sample Theory
Review of random vectors and the multivariate normal distribution
Stein’s lemma, Scheffe’s theorem, and the Glivenko-Cantelli theorem
Stochastic order symbols: O_p, o_p and basic properties
Weak convergence, continuous mapping theorem, Slutsky’s lemma
The delta method and variance stabilizing transformations
The sample correlation coefficient and the chi-squared test
Method of moments: asymptotic normality
Kullback-Liebler divergence and Fisher information
Maximum likelihood: consistency and asymptotic normality
The Cramer-Rao inequality
Limiting distribution of likelihood ratio tests
2. Other Topics
Review of Chernoff bounds, Hoeffding’s MGF and probability inequalities
Azuma-Hoeffding inequality, McDiarmid’s bounded difference inequality
Gaussian concentration for Lipschitz functions
Gaussian extreme value theorem
Review of convex sets and functions, convex hulls and extreme points
Gaussian mean width
High dimensional estimation with constraints
Symmetrization and contraction
Comparison theorems for Gaussian random vectors, Slepian’s lemma
Covering and packing numbers
Gaussian processes: Upper and lower bounds on expected maxima via chaining, minoration.
Gaussian sequence model, James-Stein estimator
1. When looking over your notes or the reading assignment, keep a pencil and scratch paper on hand, and try to work out the details of any argument that is not completely clear to you. Even if the argument is clear, it can be helpful to write it down again in a similar way, or a different way, in order to test and strengthen your understanding.
2. Always look over the notes from lecture k before attending lecture k+1. You will get much more out of the material if you can maintain a sense of continuity and keep the “big picture” in mind. This includes mathematical ideas that can make multiple appearances in slightly different forms.
3. It is important to know what you know, but it’s especially important to know what you don’t know. As you look over the reading material and your notes, ask yourself if you (really) understand it. Keep careful track of any concepts and ideas that are not clear to you, and make efforts to master these in a timely fashion. One good way of seeing if you understand an idea or concept is to write down (or state out loud) the associated definitions and basic facts, without the aid of your notes, in full, grammatical sentences. Translating ideas from mathematics to complete English sentences, and back again, is an important research skill, and a good way to assess your understanding.