Welcome to CSI 972 / STAT 972

Mathematical Statistics I

Fall, 2008

Instructor: James Gentle

Lectures: Thursday, 4:30-7:10pm, Innovation Hall, room 336

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.

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
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.
I will also use my Companion notes.
I plan to cover the material through Chapter 4 in the Companion during the fall semester for 972.
I plan to cover the remainder in 973.
See also the references listed in the general description.

One learns mathematical theory primarily by individual work; that is, by supplying the successive steps in solving a problem or proving a theorem. Some mathematical theory is learned and reinforced by passive activities such as reading or listening to lectures and discussions, and the assigned readings and weekly lectures are meant to serve this purpose. The reading assignments listed in the schedule below should be carried out with a pencil and paper in hand. The readings should be iterated as necessary to achieve a complete understanding of the material.

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. The homework assignments are long, so they should be begun long before they are due. Start each problem on a new sheet of paper and label it clearly. The problems do not need to be worked sequentially (some are much harder than others); when you are stuck on one problem, go on to the next one.

    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 28

    Course overview; notation; etc.
    How to learn mathematical statistics (working problems and remembering the big picture); "easy pieces".
    Basic math operations; methods of proving statements.
    Linear algebra
    Fundamentals of measure theory; sigma-fields, measures, integration and differentiation.
    Fundamentals of probability theory: random variables and probability distributions, and expectation; important inequalities.
    Reading assignments: Companion notes, Appendices A, B, and C, and Chapter 1, and Shao, Chapter 1.
    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 1, due September 18: In Exercises 1.6: problems 4, 5, 8, 18, 23, 43, 58, 63, 78, 90, 101, 102, 103, 127. 158

    Week 2, September 4

    Continuation of material from Week 1.

    Week 3, September 11

    Conditional expectation, joint distributions, and independence
    Asymptotic properties
    Limit theorems
    Lecture notes
    Assignment 1 comments/solutions.
    Reading assignments: Companion notes, Chapter 2, and Shao, Chapter 2.

    Week 4, September 18

    Fundamentals of statistics.
    Lecture notes
    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 16: In Exercises 2.6: problems 3, 4, 8, 20, 28, 33, 63, 81, 116, 123

    Week 5, September 25

    Decision theory, confidence sets, and hypothesis testing.
    Lecture notes

    Week 6, October 2

    Asymptotic inference
    Lecture notes
    Handout take-home portion of midterm. Due October 16.

    Week 7, October 9

    Midterm exam.
    Closed book and closed notes except for one sheet (front and back) of prewritten notes.
    Assignment 2 comments/solutions.

    Week 8, October 16

    Reading assignments: Companion notes, Chapter 3.

    Week 9, October 23

    Lecture notes
    Bayesian inference.

    Week 10, October 30

    Lecture notes
    Bayesian testing and credible regions.

    Week 11, November 6

    Lecture notes
    Reading assignments: Companion notes, Chapter 4.

    Week 12, November 13

    Lecture notes
    Maximum likelihood.

    Week 13, November 20

    Maximum likelihood.

    Week 14, December 4

    December 11

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