Class meetings, Fall 2024: Tuesdays and Thursdays 3:30pm – 4:45pm in Gardner 105
Prerequisites: MATH 233 or 235 (multivariable calculus); STOR 215/315 or MATH 381 (introduction to logic and proofs); STOR 435 (calculus based probability). A more detailed list of prerequisites is given below.
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: Mondays 1:30-2:30pm, Tuesdays 4:45-5:20pm
Teaching Assistant: Yang Xiang, xya@unc.edu
TA Office Hours: Mondays 12:30-2pm
Office: B-36 Hanes
Grader: Yanhui Ren, yar@unc.edu
Description and Audience: STOR 555 provides an introduction to theoretical statistics at the advanced undergraduate level. The course assumes knowledge of elementary statistics, multivariable calculus, discrete mathematics, and calculus based probability. The course is targeted to undergraduate students pursuing the STAN major, but may be appropriate for undergraduate and masters students in other departments with suitable mathematical and statistical background.
Goals: The course has two primary goals. The first goal is to provide students with a rigorous introduction to some of the key ideas and results in the theory of statistics. While these results can be viewed in purely mathematical terms, we will emphasize their importance and interpretation from the point of view of statistical inference and data analysis. The second goal of the course is to introduce students to some of the basic mathematical techniques underlying theoretical statistics, and to train them in their application. A detailed list of statistical and mathematical topics is given in the syllabus below.
Most topics will be self-contained, results being derived from first principles and the prerequisite material.
Protocol for lectures
- Please arrive on-time, before the beginning of class, and leave only after the end of class. If you need to arrive late or leave early, let the instructor know in advance.
- The only electronic devices permitted during lectures are laptops or tablets used to take notes. Use of cell phones is not permitted.
Texts: The primary texts for the course are
“Essentials of Statistical Inference” by G.A. Young and R.L. Smith
“Statistical Inference” by G. Casella and R.L. Berger.
Honor Code: Students are expected to adhere to the UNC honor code at all times.
Homework policy: Homework for the class will be handled using Gradescope. Homework problems will be assigned regularly throughout the semester, usually every week. Homework assignments are typically due before class, so please be prepared to submit your homework at that time. Each assignment will be graded: late/missed assignments will receive a grade of zero. All assignments will have equal weight. In computing a student’s overall score for the course, their lowest homework score will be dropped. This provision is meant to cover exceptional situations in which a student is unable to turn in an assignment due to circumstances beyond his/her control. Under normal circumstances students are expected to turn in every homework assignment.
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 homework assignments, and in-class midterm and quizzes, and an in-class final exam. A breakdown of the weights assigned to each of these items is given below.
Homework | 20% | Weekly |
Quiz 1 | 8% | September 19 |
Midterm | 24% | October 15 |
Quiz 2 | 8% | November 7 |
Final | 40% | UNC Final Exam Schedule |
Prerequisites: Students should have a good understanding of multivariate calculus, logic and proof techniques as taught in STOR 215/315 or Math 381, and probability as taught in STOR 435 or 535. Students should also have some exposure to vectors and basic matrix algebra. Specific prerequisites include the following:
- Derivatives and partial derivatives, elementary Taylor series
- Multiple integrals, change of variables
- Conditional probability and conditional expectation, joint distributions, variance and covariance, moment generating functions
- Discrete distributions, including the Bernoulli, binomial, geometric, and Poisson
- Continuous distributions, including the Normal, exponential, and Uniform
- Markov and Chebyshev inequalities
Tentative Syllabus (subject to change):
1. Background and review
- Probability models, distributions, CDFs, discrete and continuous random variables
- Basic properties of expectation, variance, and covariance
2. Review and derivation of key continuous distributions
- Gamma function, convolutions
- CDF method, change of variables
- Normal, gamma, beta, chi-squared, and t distributions
- Stein’s lemma
3. Statistical Models
- Location, scale, and location-scale families
- Exponential families
4. Decision Theory
- Framework and risk functions
- Admissability
- Bayes rules and minimax rules
5. Data Reduction
- Sufficiency and minimal sufficiency
- Jensen’s inequality
- Rao-Blackwell theorem
6. Point Estimation
- Setting and decision theoretic framework
- Method of moments
- Maximum likelihood
- Bayesian point estimates, conjugate priors
- Complete statistics, Lehmann-Scheffe
- Ancillary statistics
6. Hypothesis Testing
- Setting and decision theoretic framework
- Likelihood ratio test
- Power function
- UMP tests and the Neyman-Pearson Lemma
- p-values
- Bayes factors
6. Confidence Intervals
- Setting and decision theoretic framework
- Confidence intervals via inverting tests and pivotal quantities
7. More advanced topics (as time permits)
- Hierarchical modeling and empirical Bayes estimation
- Probability inequalities
- Asymptotic properties of maximum likelihood estimators
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 state the associated definitions and basic facts, verbally or in writing, without the aid of notes and 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:
- Authorized University activities
- Disability/religious observance/pregnancy, as required by law and approved by Accessibility Resources and Service and/or the Equal Opportunity and Compliance Office (EOC)
- 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).