Welcome to CSI 972 / IT 972

Mathematical Statistics I

Fall, 2005

Wednesday, 7:20-10:00pm, Innovation Hall, room 203

Instructor: James Gentle

If you send email to the instructor, please put "CSI 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/

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. 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. Topics include the general formulation of statistical decision theory and optimal decision rules.

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.

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)

    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.

    Week 1, August 31

    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.
    Fundamentals of measure theory: sigma-fields, measures, integration and differentiation.
    Fundamentals of probability theory: random variables and probability distributions, and expectation; important inequalities. (Shao, Sections 1.1 and 1.3)
    Reading assignment: Read Shao, Sections 1.1 through 1.3.
    Exercises assignment for discussion: In Exercises 1.6: problems 4, 8, 31, 36, 51, 63, 85
    Assignment 1a, due September 21: In Exercises 1.6: problems 12, 14, 30, 38, 53, 55, 60
    Solutions, comments.
    Assignment 1b, due September 28: In Exercises 1.6: problems 70, 78, 91, 97, 128, 161
    Solutions, comments.

    Week 2, September 7

    Review material in Sections 1.1 and 1.3. Discuss integration and differentiation. Discuss and work problems from Chapter 1.
    Reading assignment: Read Shao, Sections 1.4 and 1.5.

    Week 3, September 14

    Continue discussion of problems from previous week.
    Sequences of sets; intervals on the reals
    Continue discussion of fundamentals of probability theory: special stochastic processes, asymptotics. (Shao, through Chapter 1)
    Reading assignment: Read Shao, Sections 2.1 and 2.2.

    Week 4, September 21

    Fundamentals of statistics: distributional models, parametric classes. (Shao, through Section 2.2)
    Reading assignment: Read Shao, Sections 2.3 and 2.4.
    Assignment 2, due October 12 (but to be turned in October 19): In Exercises 2.6: problems 3, 4, 9, 19. 30, 44, 56, 66, 74, 84, 93, 101, 121
    Hints, comments.

    Week 5, September 28

    Inference. (Shao, through Section 2.4)
    Reading assignment: Read Shao, Section 2.5.

    Week 6, October 5

    Asymptotic inference. (Shao, through Chapter 2)
    Handout takehome portion of midterm (remember the username and password). Due October 19.
    Sample from a previous year.
    Solution/comments. Once you look at this, you destroy the value in the sample.

    Week 7, October 12

    Finish Chapter 2; review.

    Week 8, October 19

    Midterm exam.
    Sample from a previous year.
    Closed book and closed notes except for one sheet (front and back) of prewritten notes.
    Reading assignment: Read Shao, Sections 3.1 and 3.2.
    Assignment 3, due November 16 (or by noon November 22): In Exercises 3.6: problems 3, 6, 19, 32(a)(b)(c), 35(a)(b)(c), 44 (note U_n should be nU_n), 52, 60, 91, 106.
    Comments.

    Week 9, October 26

    Discuss inclass portion and takehome portion of midterm.
    UMVU estimation.
    Reading assignment: Read Shao, Sections 3.3 and 3.4.

    Week 10, November 2

    U statistics; least squares estimation.
    Reading assignment: Read Shao, Section 3.5.

    Week 11, November 9

    LSE in linear models.
    Some basic facts about matrices and vectors
    More information about matrices and vectors
    Reading assignment: Read Shao, Sections 4.1 and 4.2.
    Assignment 4, due December 7 (but not to turn in): In Exercises 4.6: problems 1(a), 1(b), 2(a), 2(b), 13, 14, 15, 17, 18, 19(b), 27, 30, 47, 52, 89, 91

    Week 12, November 16

    Finite population sampling; miscellaneous topics in unbiased estimation.
    Bayesian methods.
    Reading assignment: Read Shao, Section 4.3.

    (No Class November 23)

    Week 13, November 30

    Bayesian methods; invariance.

    Week 14, December 7

    Invariance; admissibility.
    Review.
    Handout takehome portion of final (due December 14).

    December 14

    7:30pm - 10:15pm Final Exam.
    Closed book and closed notes except for one sheet of prewritten notes.
    Sample from a previous year.