STAT 5380 -- Advanced Mathematical Statistics I -- Spring 2024
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 109.
Office Hours: TWR 1:00-2:00, 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 (20%), two semester tests (25% each), and a comprehensive final
exam (30%). 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:
- Test 1: Thursday Feb 22. (After completion of Homework Set 5.)
- Test 2: Thursday April 4. (After completion of Homework Set 9.)
- Final Exam: 10:30 - 1:00 p.m. on Monday May 6.
Homework Problem Sets
There will be weekly problem sets due on fridays, posted on Blackboard. 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.
- Set 1: Prove Rmk 1.1.2 (notes); Ch 1 (TPE): 6.3, 6.4, 6.6.
- Set 2: Prove Theom 1.3.15 (notes); Ch 1 (TPE): 5.1, 5.2, 5.3, 5.7, 6.7, 6.8, 6.10, 6.30.
- Set 3: Ch 2 (TPE): 1.3, 1.9, 1.17, 1.18, 2.14, 2.24, 4.2, 4.3.
- Set 4: Ch 2 (TPE): 5.8, 5.18, 5.23, 5.24, 6.8, 6.9(a), 6.11.
- Set 5: Ch 3 (TPE): 1.4, 1.10, 1.11, 3.2, 3.3, 3.7*, 3.13, 3.15, 3.22.
- Set 6: Ch 4 (TPE): 1.4, 1.6, 1.7(a)*, 1.9, 1.10, 1.11, 3.6, 3.9*, 6.10*.
- Set 7: handout.
- Set 8: handout.
- Set 9: Ch 6 (TPE): 3.4, 3.5, 3.15(c), 3.18*, 6.11, 6.14.
Note: problems with an asterisk (*) contain typographical errors, and you should refer to the above errata list for the correct versions.
Policies
- Required Texas Tech Policies can be found here.
- Recommended Texas Tech Policies can be found here.
- Use of Generative AI Tools. The use of generative AI tools (such as ChatGPT) is not permitted in this course; therefore, any use of AI tools for work in this class may be
considered a violation of Texas Tech's Academic Integrity policy and
the Student Code of Conduct since the work is not your own. The use of
unauthorized AI tools will result in referral to the Office of Student
Conduct.
- 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 specific policies 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 copy it from someone else.
- Tests: Any form of collaboration in tests, including trying to see
what the person next to you is writing, is strictly forbidden and will not be tolerated.
top of page