Welcome to CSI 972 / STAT 972

Mathematical Statistics I

Fall, 2007

Instructor: James Gentle

Lectures: Monday, 4:30-7:10pm, Enterprise Hall, room 173

Some of the lectures will be based on the instructor's notes posted on this website. Some lectures will be accompanied only by notes written on the board.

Recitations (optional): Monday, 7:30pm, Research Building I, room 92

During the optional recitation periods students and/or the instructor will discuss exercises, especially those listed as "for practice and discussion". The instructor may also discuss some of the class notes.

If you send email to the instructor, please put "CSI 972" or "STAT 972" in the subject line.

This course is part of a two-course sequence. The general description of the two courses is available at mason.gmu.edu/~jgentle/csi9723/

The prerequisites for this course include a course in mathematical statistics at the advanced calculus level, for example, at George Mason, CSI 672 / STAT 652, "Statistical Inference".

This course is primarily on the theory of estimation. It begins with a brief discussion of probability theory, and then covers fundamentals of statistical inference. The principles of estimation are then explored systematically, beginning with a general formulation of statistical decision theory and optimal decision rules. Minimum variance unbiased estimation is covered in detail. Topics include sufficiency and completeness of statistics, Fisher information, bounds on variances, consistency and other asymptotic properties. Other topics and approaches in parametric estimation are covered in detail.

The text is Jun Shao (2003), Mathematical Statistics, second edition, Springer.
Be sure to get the corrections at the author's website
I plan to cover the material through Section 4.3 in 972.
I plan to cover the remainder in 973.
A useful supplement is Jun Shao (2005), Mathematical Statistics: Exercises and Solutions, Springer. My assigned "exercises for practice and discussion" are all solved (or at least partially solved) in this book.
See also the references listed in the general description.

Student work in the course (and the relative weighting of this work in the overall grade) will consist of

  • homework assignments (25)
  • a midterm consisting of an in-class component and a take-home component (30)
  • a final exam consisting of an in-class component and a take-home component (45)

    Each homework will be graded based on 100 points, and 5 points will be deducted for each day that the homework is late.

    Each student enrolled in this course must assume the responsibilities of an active participant in GMU's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. The GMU policy on academic conduct will be followed in this course.

    Students are free to discuss the homework with each other or anyone else, and are free to use any reference sources. Explicit copying should not be done.

    Students are not to communicate concerning exams with each other or with any person other than the instructor. On take-home exams, any passive reference is permissible (that is, the student cannot ask someone for information, but the student may use any existing information from whatever source).

    For in-class exams, one sheet of notes will be allowed.

    An approximate schedule is shown below. As the semester progresses, more details will be provided, and there may be some slight adjustments.
    Notes are posted in a password-protected directory.
    Students are expected to read the relevant material in the text prior to each class (after the first one).
    Students are strongly encouraged to solve the "exercises for practice and discussion", especially those marked with an asterisk.

    Week 1, August 27

    Course overview; notation; etc.
    How to learn mathematical statistics (working problems and remembering the big picture); "easy pieces".
    Basic math operations ( notes); methods of proving statements.
    Linear algebra ( notes).
    Fundamentals of measure theory ( notes): sigma-fields, measures, integration and differentiation.
    Reading assignment: Read Shao, Sections 1.1 through 1.3.
    Exercises for practice and discussion: In Exercises 1.6: problems 12, 14, 30, 31, 36, 38, 51, 53, 55, 60, 70, 85, 91, 97, 128, 161
    Assignment 1a, due September 17: In Exercises 1.6: problems 4, 5, 8, 18, 23, 43, 58, 63
    Assignment 1b, due September 24 (delayed to October 1): In Exercises 1.6: problems 78, 90, 101, 102, 103, 127, 158

    (No class on September 3)

    Week 2, September 10

    Fundamentals of probability theory ( notes): random variables and probability distributions, and expectation; important inequalities.
    Reading assignment: Read Shao, Sections 1.4 and 1.5.

    Week 3, September 17

    Conditional expectation, joint distributions, and independence ( notes).
    Asymptotic properties ( notes).
    Limit theorems ( notes).
    Assignment 1a comments/solutions.
    Reading assignment: Read Shao, Sections 2.1 and 2.2.

    Week 4, September 24

    Shao, Chapter 2: Fundamentals of statistics:distributional models, parametric classes ( notes).
    Decision theoretic approach ( notes).
    Probability statements for inference ( notes).
    Reading assignment: Read Shao, Sections 2.3 and 2.4.
    Exercises assignment for practice and discussion: In Exercises 2.6: problems 9, 13, 19, 23, 25, 30, 44, 56, 66, 74, 84, 93, 101, 115, 121
    Assignment 2, due October 22: In Exercises 2.6: problems 3, 4, 8, 20, 28, 33, 63, 81, 116, 123
    Comments/hints.

    Week 5, October 1

    Decision theory, confidence sets, and hypothesis testing.
    There will be no separate recitation session.
    Assignment 1b comments/solutions.
    Reading assignment: Read Shao, Section 2.5.

    Week 6, October 9 (Tuesday)

    Shao, Chapter 2: Asymptotic inference ( notes).
    Class will end at 6:45pm in order for the instructor to attend the special lecture "What Happened Before the Big Bang? A Novel Answer to a Profound Cosmological Puzzle" by Roger Penrose. Students are also invited to attend this lecture in Dewberry Hall in the Johnson Center.
    Because of this lecture, there will be no recitation session this evening.

    Week 7, October 15

    Shao, Chapters 1 and 2. Review.
    Handout take-home portion of midterm. Due October 29.
    Sample from a previous year.

    Week 8, October 22

    Midterm exam.
    Sample from a previous year.
    Closed book and closed notes except for one sheet (front and back) of prewritten notes.
    Assignment 2 omments/hints.
    Reading assignment: Read Shao, Sections 3.1 and 3.2.
    Exercises assignment for practice and discussion: In Exercises 3.6: problems 6, 19, 33, 34*, 60, 70, 106, 107 (note typo), 111.
    Assignment 3, due November 19: In Exercises 3.6: problems 3, 16, 32(a)(b)(c), 35(a)(b)(c), 44 (note U_n should be nU_n), 52, 91, 109, 114.
    Comments/hints.

    Week 9, October 29

    Take-home exam due.
    Discuss in-class midterm.
    Shao, Chapter 3: UMVU estimation (notes).
    Reading assignment: Read Shao, Sections 3.3 and 3.4.

    Week 10, November 5

    Discuss take-home midterm.
    Shao, Chapter 3: U statistics (notes); least squares estimation (notes).
    Reading assignment: Read Shao, Section 3.5.

    Week 11, November 12

    Shao, Chapter 3: LSE in linear models; finite population sampling ( notes); miscellaneous topics in unbiased estimation ( notes).
    Reading assignment: Read Shao, Sections 4.1 and 4.2.
    Exercises assignment for practice and discussion: In Exercises 4.6: problems 2(a)(b), 13, 14, 15, 19(b), 27, 30, 47, 52, 89, 91
    Assignment 4, due December 3 (but can turn in December 17): In Exercises 4.6: problems 1(a)(b), 17, 18, 32(a), 38(a)(b), 57(a)
    Comments/hints.

    Week 12, November 19

    Shao, Chapter 4:
    Invariance, equivariance ( notes).
    Bayesian methods ( notes).
    Assignment 3 comments/solutions.
    Reading assignment: Read Shao, Section 4.3.

    Week 13, November 26

    Shao, Chapter 4: Bayesian methods; MCMC ( notes).

    Week 14, December 3

    Shao, Chapter 4: More on Bayesian methods.
    Admissibility ( notes).
    Takehome portion of final (due December 17).
    I put the exam up on Tuesday, but I forgot to put the link to it in this file until Wednesday! Remember, if you find a typo, or something that might be a typo, let me know immediately.
    Review during recitation session.

    Week 15, December 10

    No official class.
    I will be in the regular lecture room at the regular time, however, for anyone who wants to show up for questions and discussions.

    December 17

    4:30pm - 7:15pm Final Exam.
    Closed book and closed notes except for one sheet of prewritten notes.