**Class meetings, Spring 2017:** Tuesday and Thursday 11:00am – 12:15pm 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: nobel@email.unc.edu Phone: 919-962-1352.

**Office Hours:** Monday 2:10-3:25

**TA:** Xi Chen

**Office:** Hanes B48 Email: xich@live.unc.edu

**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 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.

Homework | 15% |

Midterm | 35% |

Final | 50% |

Note: The final exam will be given at the time and date specified by the UNC Final Exam Schedule.

**Other sources:**

“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.

**Tentative Syllabus:**

**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

*Asymptotic efficiency

**2. Other Topics
**

Review of Chernoff bounds, Hoeffding’s MGF and probability inequalities

Azuma-Hoeffding inequality, McDiarmids 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

**Study tips: **

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.

2. 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.

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.