Welcome to CSI 972 / STAT 972

Mathematical Statistics I

Fall, 2011

Instructor: James Gentle

Lectures: Tuesday, 4:30-7:10pm, Robinson Hall, room B203

Some of the lectures will be based on the instructor's notes posted on this website. The lectures themselves will not be posted. Some lectures will be accompanied only by notes written on the board.

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.


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".

Text and other reading materials

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 plan to cover most of the material in the first four chapters in Shao during 972 in the fall semester, and I plan to cover the most of the remainder in 973.
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 of the course.
My evolving Companion notes may also be useful. These notes, which include an index and a bibliography, are not complete, and are not meant to be. Their purpose is to provide a few additional examples, and some more detailed discussion of some things. I will add to them frequently, so I do not recommend printing them.

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.


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.


    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 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).


    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 30

    Course overview; notation; etc.
    How to learn mathematical statistics (working problems and remembering the big picture); "easy pieces".
    Fundamentals of measure theory: sigma-fields, measures, integration and differentiation.
    Reading assignments: Companion notes, Sections 0.0 and 0.1 and Chapter 1, and Shao, Chapter 1.
    Exercises for practice and discussion: In Shao Exercises 1.6: problems 12, 14, 30, 31, 36, 38, 51, 53, 55, 60, 70, 85, 91, 97, 128, 161.
    Assignment 1, due September 6: In Shao Exercises 1.6: problems 2 (this is just the definition that I give -- but you should use Shao's informal definition as the "smallest", and also you must prove that it is nonempty), 6, 17.
    In Companion, Exercises 0.0.14, 0.1.1, 0.1.4, 0.1.8, 0.1.10, 0.1.13, 0.1.25.

    Week 2, September 6

    Fundamentals of probability theory: random variables and probability distributions, and expectation; important inequalities.
    Assignment 2, due September 13: In Shao Exercises 1.6: problems 41, 58, 63.
    In Companion, Exercises 1.8, 1.12, 1.24, 1.29, 1.39, 1.46, 1.56(a).

    Week 3, September 13

    Probability theory and families of probability distributions.
    Assignment 3, due September 20: In Shao Exercises 1.6: problems 78, 99, 102, 105, 127, 159.
    In Companion, Exercises 1.67, 1.69, 1.73, 1.74

    Week 4, September 20

    Families of probability distributions useful in statistical applications.
    Basic statistical concepts: Sufficiency and completeness.

    Reading assignments: Companion notes, Chapter 2, and Shao, Chapter 2.
    Exercises for practice and discussion: In Shao Exercises 2.6: problems 9, 13, 19, 23, 25, 30, 44, 56, 66, 74, 84, 93, 101, 115, 121.
    Assignment 4, due September 27: In Shao Exercises 2.6: problems 3, 5, 7, 20, 27.

    Week 5, September 27

    Basic statistical concepts: Decision theory, confidence sets, and hypothesis testing.

    Week 6, October 4

    In class midterm exam. Closed book and closed notes except for one sheet (front and back) of prewritten notes.
    Due October 18
    Between now and the end of class on October 18, students are not to discuss homework or other aspects of the course (including the takehome of course!) with anyone other than the instructor.

    October 11

    Class does not meet this week

    Week 7, October 18

    Takehome midterm exam due.
    Review inclass exam.
    Decision theory.
    Assignment 5, due October 25: In Shao Exercises 2.6: problems 28, 33, 63, 81, 98, 116, 123, 127.

    Week 8, October 25

    Decision theory and general review of Shao Chapter 2.
    Bayesian inference

    Reading assignment: Companion notes, Chapter 3.
    Assignment 6, due November 1: In Shao problems 4.1(a)(b), 4.17, 4.18, 4.31, 4.32(a), 4.38(a)(b).

    Week 9, November 1

    Bayesian inference
    Assignment 7, due November 8: In Companion, Exercises 3.7, 3.9, 3.11, 3.12, 3.14, 3.15 (Exercises 3.2, 3.3, 3.5, 3.7, 3.8, 3.9 in version of Companion dated prior to November 8)

    Week 10, November 8

    Bayesian inference
    Assignment 8, due November 15: In Companion, Exercises 3.16, 3.20, 3.21, 3.23, 3.24 (Exercises 3.12, 3.14, 3.15, 3.17, 3.18 in version of Companion dated between November 8 and November 18)

    Week 11, November 15

    Unbiased estimation.
    Assignment 9, due November 22: In Shao problems 3.9, 3.15, 3.32(a)(c)(g), 3.35(a)(b)(c).

    Week 12, November 22

    Unbiased estimation.
    U statistics.
    Linear models.
    Finite population sampling.
    Assignment 10, due December 7: In Shao problems 3.43, 3.45, 3.46, 3.52, 3.61, 3.91.

    Week 13, November 29

    Unbiased estimation.
    Asymptotically unbiased estimators.
    Assignment 11, due December 7: In Shao problems 3.106, 3.107, 3.109, 3.111, 3.114.

    Week 14, December 7

    Miscellaneous topics in estimation.

    December 13

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