## STAT 5380 -- Advanced Mathematical Statistics I -- Spring 2018

### Basic Information

Course instructor: Dr. Alex Trindade, 228 Mathematics & Statistics Building.
Course Meets: MWF 1:00-1:50 in MATH 010.
Office Hours: MWF 9-12 in 201F, or by appointment.

### Text Book

Required (available as e-book through TTU library):
• Theory of Point Estimation (TPE), by Lehmann and Casella, 2nd edition, 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, Springer. (Companion to TPE used in STAT 5381.)
• 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, by Bickel and Doksum, 2nd ed., 2001. (Similar content and level as TPE and TSH.)
• A Probability Path, by Resnick, 1999, Birkhauser. (For reviewing the measure theory.)
• Elements of Large Sample Theory, by Lehmann, 2nd ed., 1999, Springer. (Detailed coverage of large sample 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. 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 Lehmann (1999), or Brockwell & Davis (1991, Ch. 6))
• Maximum likelihood estimation (TPE: ch. 6)
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: March 2. (After completion of Homework Set 5.)
• Test 2: April 13. (After completion of Homework Set 9.)
• Final Exam: 1:30 - 4:00 p.m. on Thursday May 10.

### Homework Problem Sets

There will be weekly problem sets due on thursdays. You are expected to do all the assigned problems, but only a subset may actually be graded. Start each problem on a separate page. All work handed in must be stapled together. No late submissions will be accepted.

• Set 1 (Fri Jan 26): Prove Rmk 1.1.2 (notes); Ch 1 (TPE): 6.3, 6.4, 6.6.
• Set 2 (Fri Feb 2): Ch 1 (TPE): 6.7, 6.8, 6.10, 6.11, 6.12, 6.13.
• Set 3 (Fri Feb 9): Prove Theom 1.3.15 (notes); Ch 1 (TPE): 5.1, 5.2, 5.3, 5.7.
• Set 4 (Fri Feb 16): Ch 2 (TPE): 1.1, 1.2, 1.3, 1.9.
• Set 5 (Fri Feb 23): Ch 2 (TPE): 1.14, 1.17, 1.18, 2.14, 2.24.
• Set 6 (Fri Mar 9): Ch 2 (TPE): 4.2, 4.3, 5.6, 5.8, 5.18, 5.23, 5.24.
• Set 7 (Fri Mar 23): Ch 2 (TPE): 6.8, 6.9(a), 6.11; Ch 3 (TPE): 1.1, 1.4, 1.7, 1.10, 1.11.
• Set 8 (Fri Mar 30): Ch 3 (TPE): 1.13, 1.18, 2.18, 2.19*, 3.2, 3.3.
• Set 9 (Fri Apr 6): Ch 3 (TPE): 3.7*, 3.13, 3.15, 3.22; Ch 4 (TPE): 1.4, 1.6, 1.7(a)*.
• Set 10 (Fri Apr 20): handout.
• Set 11 (Fri Apr 27): handout.
• Set 12 (Fri May 4): Ch 6 (TPE): 2.8, 3.4, 3.5, 3.15(c), 3.18*, 4.4*.

Note: problems with an asterisk (*) contain typographical errors, and you should refer to the above errata list for the correct versions.

### Policies

• Class Attendance. Your attendance alone will not impact your grade, but missing exams and assignments will. Whether an absence is excused or unexcused is determined solely by me, with the exception of absences due to religious observance and officially approved trips (see below).
• Make-up Exams: These may be granted in exceptional circumstances if you provide me with a valid excuse (such as a note from a physician, an obituary, etc.).
• Absence due to religious observance: The Texas Tech University Catalog states that a student shall be excused from attending classes or other required activities, including examinations, for the observance of a religious holy day, including travel for that purpose. A student who intends to observe a religious holy day should make that intention known in writing to the instructor prior to the absence. A student who is absent from classes for the observance of a religious holy day shall be allowed to take an examination or complete an assignment scheduled for that day within a reasonable time after the absence.
• Absence due to officially approved trips: The Texas Tech University Catalog states that the department chairpersons, directors, or others responsible for a student representing the university on officially approved trips should notify the student's instructors of the departure and return schedules in advance of the trip. The instructor so notified must not penalize the student, although the student is responsible for material missed. Students absent because of university business must be given the same privileges as other students.
• Illness and Death Notification. The Center for Campus Life is responsible for notifying the campus community of student illnesses, immediate family deaths and/or student death. Generally, in cases of student illness or immediate family deaths, the notification to the appropriate campus community members occur when a student is absent from class for four (4) consecutive days with appropriate verification. It is always the student's responsibility for missed class assignments and/or course work during their absence. The student is encouraged to contact the faculty member immediately regarding the absences and to provide verification afterwards. The notification from the Center for Campus Life does not excuse a student from class, assignments, and/or any other course requirements. The notification is provided as a courtesy.
• Students with Disabilities. Any student who because of a disability may require special arrangements in order to meet course requirements should contact the instructor as soon as possible to make any necessary accommodations. Student should present appropriate verification from AccessTECH. No requirement exists that accommodations be made prior to completion of this approved university procedure.
• Civility in the Classroom. It is expected that everyone will behave in a manner that is conducive to learning. One common disruption is cell phones. Please turn these off in class.
• Academic Integrity. Is assumed and expected at all times. Students are advised to acquaint themselves with the Code of Student Conduct.

It is the aim of the faculty of Texas Tech University to foster a spirit of complete honesty and a high standard of integrity. The attempt of students to present as their own any work that they have not honestly performed is regarded by the faculty and administration as a serious offense and renders the offenders liable to serious consequences, possibly suspension.

• 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.

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