Welcome to CSI 972 / STAT 972

Mathematical Statistics I

Fall, 2012

Instructor: James Gentle

Lectures: Tuesday, 4:30-7:10pm, Innovation Hall, room 133

The lectures will not be printed or posted on the web.

Course Description

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 begins with a brief discussion of measure theory and probability theory. Next, it 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. Bayesian decision rules are then considered in some detail. 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 addressed.

Prerequisites

The prerequisites for the first course include a course in mathematical statistics at the advanced calculus level, for example, at George Mason, CSI 672 / STAT 652, "Statistical Inference", and a measure-theory-based course in probability, for example, at George Mason, CSI 971 / STAT 971, "Probability Theory".

The required text is The Theory of Point Estimation, second edition, 1998 by E.L. Lehmann and George Casella, ISBN # 0-387-98502-6. I will refer to this text as TPE2.
The book has a few errata, mostly typos. (There were a few additional ones in the first printing, but I assume that by now most people will have the second printing or later.)
At the level of this course, no single text can cover "everything". The student is encouraged to study other texts on the various topics; see, for example, 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.

Email Communication

The primary means of communication outside of class will be by email.

Students must use their Mason email accounts to receive important University information, including messages related to this class. (You may, of course, foward email from your Mason email account to one that you check regularly.)

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

Grading

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 (20 each)
  • a final in-class exam (35)

    For in-class exams, one sheet of notes will be allowed. The preparation of that sheet is one of the most important learning activities.

    Homework

    Each homework will be graded based on 100 points, and 5 points will be deducted for each day that the homework is late. No homework will be accepted after it is more than 3 days late.
    The homework assignments are long, so they should be begun well 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.
    Homework will not be accepted as computer files; it must be submitted on paper.

    Academic honor

    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.

    Collaborative work

    Except during a period in which a take-home exam is being worked on, students are free to discuss homework problems or other topics with each other or anyone else, and are free to use any reference sources. Group work and discussion outside of class is encouraged, but of course explicit copying of homework solutions 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).

    Schedule

    An approximate schedule is shown below. As the semester progresses, more details may be provided, and there may be some slight adjustments.

    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; preparations (TPE2, Chapter 1) Reading assignment: TPE2, Chapter 1.
    Assignment 1, due September 4: In TPE2 Section 1.9 (page 62ff): problems 1.2, 1.3(a), 1.4(b), 1.7(b), 1.11(a),(b), 2.3, 2.4(a),(b), 3.8, 4.1, 4.13(a),(b)
    Assignment 2, due September 11: In TPE2 Section 1.9: problems 5.6(a),(b), 5.12, 5.13, 5.22, 6.16, 6.18, 7.8, 7.14(a), 8.1(b), 8.11(b), 8.21(a),(b),(c),(d)

    Week 2, September 4

    Preparations (continuation)

    Notes

    Unbiased estimation (TPE2, Chapter 2):
    UMVUE
    Reading assignment: TPE2, Chapter 2.
    Assignment 3, due September 11 (or September 18): In TPE2 Section 2.7 (page 129ff): problems 1.3(a),(b), 1.4(a),(b), 1.10(a),(b),(c),(d),(e), 1.13, 1.18(a),(b), 1.20, 2.1(a),(b),(c), 2.26
    Assignment 4, due September 18: In TPE2 Section 2.7: problems 2.27(a),(b), 3.23, 4.1(a), 4.6, 5.3, 5.7(a),(b),(c),6.2(a),(b),(c), 6.10(a)

    Week 3, September 11

    Unbiased estimation (continuation of Chapter 2):
    Nonparametric estimation; the information inequality

    Week 4, September 18

    Unbiased estimation (miscellaneous topics; review as necessary)

    Week 5, September 25

    Review unbiased estimation as necessary.
    Equivariance (TPE2, Chapter 3):
    First examples, location-scale families

    Reading assignment: TPE2, Chapter 3.
    Assignment 5, due October 23: In TPE2 Section 3.8 (page 207ff): problems 1.3, 1.5, 1.6, 1.13, 1.15, 1.16, 1.22, 2.1, 2.10

    Week 6, October 2

    In class midterm exam. Closed book and closed notes except for one sheet (front and back) of prewritten notes.
    Hand out midterm takehome. Due October 16
    Between now and the end of class on October 16, students are not to discuss homework or other aspects of the course (including the takehome of course!) with anyone other than the instructor.

    October 9

    Class does not meet this week

    Week 7, October 16

    Takehome midterm exam due.
    Review inclass exam.
    Equivariance (continuation of Chapter 3)

    Week 8, October 23

    Bayesian theory and methods (TPE2, Chapter 4):
    Assignment 6, due November 13: In TPE2 Section 4.8 (page 282ff): problems 1.8, 1.9, 1.10, 2.2, 2.4, 2.15 note typo in (a); the priors should be N(1/n,1), 3.9, 3.12

    October 30

    Class cancelled this week due to weather

    Week 9, November 6

    Bayesian theory and methods (continuation of Chapter 4):
    Assignment 7, due November 20: In TPE2 Section 4.8 (page 282ff): problems 4.1, 4.2, 4.3, 4.5(a), 4.7 only for 4.5(a), 4.10(a), 5.4, 5.8, 5.9, 5.10

    Week 10, November 13

    Bayesian theory and methods (continuation of Chapter 4).
    Minimaxity and admissibility (TPE2, Chapter 5).
    Assignment 8, due November 20 (can turn in late without penalty): In TPE2 Section 4.8 (page 282ff): problems 6.1, 6.2, 6.8, 7.1, 7.6. In TPE2 Section 5.8 (page 389): problems 1.1, 1.2.

    Week 11, November 20

    Minimaxity and admissibility (continuation of Chapter 5):
    Assignment 9, due December 4: In TPE2 Section 5.8: problems 1.21, 2.2, 2.23, 3.3, 3.14, 4.5, 4.7, 4.9, 4.14

    Week 12, November 27

    Minimaxity and admissibility

    Assignment 10, due December 4: In TPE2 Section 5.8: problems 5.1, 5.2, 5.5, 5.9, 6.7, 6.9, 7.1

    Week 13, December 4

    Minimaxity and admissibility

    Week 14, December 11

    Miscellaneous topics in estimation; review.

    December 18

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