STAT 5380 -- Advanced Mathematical Statistics I -- Spring 2026
Basic Information
Course instructor:
Dr. Alex
Trindade, 233 Mathematics & Statistics Building.
E-mail: alex.trindade"at"ttu.edu; Phone: 834-6164.
Course Meets: TR 3:30 in Math 010.
Office Hours: TR 2:00-3:00, F 4:00-5:00, or by appointment.
Text Books
Required (available as e-books through TTU library):
- Theory of Point Estimation (TPE), by
Lehmann and Casella, 2nd ed. (1998), Springer. (Can also be found on web, try to get corrected printing, 2003, or later.) Here is an errata list.
- Testing Statistical Hypotheses (TSH), by
Lehmann and Romano, 3rd ed. (2005) or 4th ed. (2022), Springer.
Helpful:
- Theoretical Statistics, by Keener, 2010, Springer. (Combines essentials of TPE and TSH in one.)
- Theory of Statistics, by Schervish, 1995, Springer. (Emphasis on Bayes inference.)
- Mathematical Statistics, by Shao, 1999, Springer. (Similar content and level as TPE and TSH.)
- Mathematical Statistics (Vol. 1), by Bickel and Doksum, 2nd ed., 2001. (Similar content and level as TPE and TSH.)
- Asymptotic Statistics, by Van der Vaart, 1998, Cambridge. (Detailed modern coverage of large sample theory.)
- A Probability Path, by Resnick, 1999, Birkhauser. (For reviewing the measure theory.)
- Time Series: Theory and Methods, by Brockwell and
Davis, 2nd ed., 1991, Springer. (See Ch. 6 for a concise coverage of large sample theory.)
- Computer Age Statistical Inference: Algorithms, Evidence, and Data Science, by
Efron & Hastie, 2016, Cambridge. (Insightful "bedside" reading.)
Course Objectives, Syllabus, and Notes
STAT 5380 aims to provide a solid theoretical foundation for statistical inference (estimation and testing). It employs a probabilistic and measure theoretic approach to formulate and solve
statistical inference problems. Material to be covered: sufficiency, decision theoretic statistical inference (minimax estimation, Bayes estimation, admissibility, shrinkage & bigdata, etc.), UMVUE, equivariance, information theoretic inference,
large sample theory, asymptotic properties of maximum-likelihood methods, and optimal hypothesis tests (UMP, UMPU).
This coverage corresponds to the following core topics:
- Preliminaries (TPE: ch. 1, TSH: chs. 1 & 2)
- Unbiasedness (TPE: ch. 2)
- Equivariance (TPE: ch. 3)
- Bayes estimation (TPE: chs. 4 & 5)
- Large sample theory (old TPE ch.5, or Van der Vaart (1998), or Brockwell &
Davis (1991, Ch. 6))
- Maximum likelihood estimation (TPE: ch. 6)
- Optimal and near-optimal tests (TSH: chs. 3 & 4)
Prerequisites: STAT 5329 (Intermediate Mathematical Statistics II) and MATH 5382 (Advanced Probability I).
A detailed set of class notes covering the core is available here.
Expected Student Learning Outcomes
Students will learn the theory behind
various statistical estimation and inference techniques, and will be capable of providing a solid theoretical justification and understanding for common statistical methods used in practice. Students can expect to spend several hours per week outside of instructional time on reading, homework, and exam preparation.
Methods of Assessing the Expected Learning Outcomes
The expected learning outcomes for the course will be assessed through:
homework sets, semester tests, and a final exam. The course grade will
be determined from homework problem sets (30%), a midsemester test (30%), and a comprehensive final
exam (40%). The traditional grading scale will be used:
- A: 90-100%.
- B: 80-89%.
- C: 70-79%.
- D: 60-69%.
- F: 0-59%.
The test schedule is as follows:
- Midterm: Thursday Mar 12. (After completion of Homework Set 8.)
- Final Exam: 7:30 - 10:00 a.m. on Thursday May 7.
Homework Problem Sets
There will be weekly problem sets due on saturdays, posted on Canvas. You are expected to do all
the assigned problems, but only a subset may actually be graded. Start each problem on a
separate page. No late submissions will be accepted.
Note: problems with an asterisk (*) contain typographical errors, and you should refer to the above errata list for the correct versions.
Policies
- Required/Recommended Texas Tech Policies can be found on the
Canvas course page.
- Use of Generative AI Tools. The use of generative AI (GAI) for working assignments is not forbidden, but it should be used with discretion. I view GAI as another way to collaborate with peers on solving problems. This can used incorrectly, e.g. when you simply copy your smart friends work (or ask GAI to solve the problem directly), or can be used correctly, e.g., when you discuss with friends the results/steps needed to find a solution after you have put some thought into it (or use GAI more like a search engine to query side questions and digest literature). Remember: you do NOT want to surrender your thought process to the "tool" (friend or GAI); the "tool" will not be available during tests!
- Electronic Devices in Tests. In the spirit of keeping costs down, I will permit the usage of apps on smart devices (phones, tablets, laptops, etc.), but any kind of communication or accessing of the web via these devices is forbidden.
- Collaboration. My policies on this are as follows.
- Homeworks: Discussion with peers regarding material/concepts covered in the
course is permitted, and is encouraged since it usually leads to greater comprehension. However, each person must write up his/her own
solution to a particular problem, and not simply have someone else do it for them.
- Tests: Any form of collaboration on tests, including e-device communication or trying to see what the person next to you is writing, is strictly forbidden and will not be tolerated.
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