Class meetings, Spring 2024: Mondays and Wednesdays 11:15am – 12:30pm in Hanes 125

Prerequisites:  STOR 654 (Theoretical Statistics I), STOR 634 (Introduction to Measure Theoretic Probability), real analysis, and linear algebra.

Registration: Enrollment and registration for the course is handled online.  Please contact Ms. Christine Keat (crikeat@email.unc.edu) if you have questions.

Instructor:  Andrew B. Nobel

Office: Hanes 308   Email: nobel@email.unc.edu

Nobel Office Hours: Tuesdays 4:00-5:15pm (via Zoom)

Teaching Assistant: Kyung-Rok Kim

TA Office Hours: Wednesday 12:30-1:30

Office: Hanes B-48  Email: kkrok@unc.edu

Audience: STOR 655 is the second course in the graduate-level 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 background.

Goals: The goal of the course is to introduce students to a number of of the key results in theoretical statistics, and to familiarize them with the mathematical techniques that underly these results.  Results covered in the course include uniform laws of large numbers, 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.  Mathematical techniques include weak convergence, the delta method, Taylor series, use of Jensen and related inequalities, Gaussian integration by parts, and symmetrization.

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.

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.

“Estimation in high dimensions: a geometric perspective” by R. Vershynin.  Lecture notes on mean width and estimation from Gaussian projections.  Link

“Gaussian estimation: Sequence and wavelet models” by I. Johnstone.  Latest version available online here.

Honor Code: Students are expected to adhere to the UNC honor code at all times.

 

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 and Exams: 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.

Prerequisites: Students should have a good understanding of the material in STOR 654 as well as a solid understanding of basic real analysis and linear algebra.  In particular, students should be familiar with the following material

• Basic theoretical statistics, including point estimation and hypothesis testing, Bernoulli, binomial, Poisson, normal, exponential, and uniform distributions, conditional distributions, variance and covariance, correlation, Markov and Chebyshev inequalities, moment generating functions
• Linear and matrix algebra, including norms, inner products, eigenvalues and eigenvectors, rank, inverse, projections, orthogonal matrices, and non-negative definite matrices
• Basic theoretical probability, including probability spaces, measurable functions, basic properties of the Lebesgue integral, regular and conditional expectations, almost sure and in-probability convergence
• Advanced calculus, including suprema and infima, limsups, liminfs, limits, open, closed, and compact sets, continuous functions, basic properties of derivatives and integrals, multivariate differentiation, multivariate integration

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 their 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
• *Asymptotic efficiency
• *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

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.  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.

Honor Code Policy

As a condition of joining the Carolina community, Carolina students pledge “not to lie, cheat, or steal” and to hold themselves, as members of the Carolina community, to a high standard of academic and non-academic conduct while both on and off Carolina’s campus. This commitment to academic integrity, ethical behavior, personal responsibility and civil discourse exemplifies the “Carolina Way,” and this commitment is codified in both the University’s Honor Code and in other University student conduct-related policies.

Accessibility Resources

The University of North Carolina at Chapel Hill facilitates the implementation of reasonable accommodations, including resources and services, for students with disabilities, chronic medical conditions, a temporary disability or pregnancy complications resulting in barriers to fully accessing University courses, programs and activities.

Accommodations are determined through the Office of Accessibility Resources and Service (ARS) for individuals with documented qualifying disabilities in accordance with applicable state and federal laws. See the ARS Website for contact information: https://ars.unc.edu or email ars@unc.edu.

Counseling and Psychological Resources

CAPS is strongly committed to addressing the mental health needs of a diverse student body through timely access to consultation and connection to clinically appropriate services, whether for short or long-term needs. Go to their website: https://caps.unc.edu/ or visit their facilities on the third floor of the Campus Health Services building for a walk-in evaluation to learn more.

Title IX Resources

Any student who is impacted by discrimination, harassment, interpersonal (relationship) violence, sexual violence, sexual exploitation, or stalking is encouraged to seek resources on campus or in the community. Please contact the Director of Title IX Compliance (Adrienne Allison – Adrienne.allison@unc.edu), Report and Response Coordinators in the Equal Opportunity and Compliance Office (reportandresponse@unc.edu), Counseling and Psychological Services (confidential), or the Gender Violence Services Coordinators (gvsc@unc.edu; confidential) to discuss your specific needs. Additional resources are available at safe.unc.edu.

University Attendance Policy

No right or privilege exists that permits a student to be absent from any class meetings, except for these University Approved Absences:

1. Authorized University activities
2. Disability/religious observance/pregnancy, as required by law and approved by Accessibility Resources and Service and/or the Equal Opportunity and Compliance Office (EOC)
3. Significant health condition and/or personal/family emergency as approved by the Office of the Dean of Students, Gender Violence Service Coordinators, and/or the Equal Opportunity and Compliance Office (EOC).