Welcome to CSI 972 / IT 972

Mathematical Statistics I

Fall, 2003

Wednesday, 7:20-10:00pm, Innovation Hall, room 316

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 www.scs.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 are covered in detail. Topics include sufficiency and completeness of statistics, Fisher information, bounds on variances, asymptotic consistency and distributions. 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.

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

a number of small assignments, problems, etc. (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)

An approximate schedule is shown below. As the semester progresses, more details will be provided on the topics to be covered, and there may be some slight adjustments. Occasionally, I will post some ``follow-up lecture notes''.

Week 1, Aug 27
Preparations: Shao, Chapter 1, fundamentals of probability; software for symbolic computations; Monte Carlo applications in mathematical statistics. Shao's Chapter 1 on probability follows Billingsley (1995) very closely, both in notation and in content. The content of course is a subset, and occasionally it is presented in a slightly different order.
Follow-up lecture notes.
Assignment 1 (due Sep 17): Exercises 1.6, beginning on page 74:
4, 8, 18, 23, 31, 36, 43, 51, 63, 78, 101, 102, 103, 127, 158.
Week 2, Sep 3
Continuation of week 1 material.
Week 3, Sep 10
Completion of week 1 material: inequalities involving the characteristic function; Lindeberg's central limit theorem.
Fundamentals of statistics (Shao, Chapter 2)
Assignment 2 (due Oct 1): Exercises 2.6, beginning on page 142:
3, 4, 5, 6, 7, 8, 9, 20, 28, 33, 36, 39, 56, 63, 74, 93, 116, 123
Week 4, Sep 17
Continuation of week 3 material. Statistical decision theory. Optimality of statistical procedures.
Week 5, Sep 24
Continuation of week 3 material.
Week 6, Oct 1
Optimality in statistical estimation under the restriction of unbiasedness: UMVUE (Shao, Section 3.1).
Follow-up lecture notes.
Assignment 3 (due Oct 15, but not to turn in): Exercises 3.6, beginning on page 217:
2, 5, 24, 32(a), 32(b), 33(a), 40
Week 7, Oct 8
Continuation of week 6 material and review.
Handout take-home portion of midterm.
Week 8, Oct 15
In-class portion of midterm
Take-home portion of midterm due.
Week 9, Oct 22
Review exam.
General topics in unbiased estimation (Shao, Sections 3.2--3.5).
Assignment 4 (due Nov 5): Exercises 3.6, beginning on page 217:
3, 6 (a),(b),(c), 19, 35 (a),(b),(c), 44, 48, 52, 53, 60, 66, 74,90, 91, 106.
Week 10, Oct 29
Continuation of week 9 material.
Week 11, Nov 5
Optimality in statistical estimation over the parameter space; Bayesian averaging. (Shao, Chapter 4).
Assignment 5 (due Dec 3, but not to turn in): Exercises 4.6, beginning on page 299:
1(a), 1(b), 2(a), 2(b), 13, 14, 17, 18, 19(b), 27, 30, 47, 89.
Week 12, Nov 12
Class does not meet.
Read Shao Section 4.2, and handout on invariance.
Week 13, Nov 19
Continuation of week 11 material. Optimality in statistical estimation over the parameter space; the minimax criterion.
Nov 26, No class (Thanksgiving holiday).
Week 14, Dec 3
Continuation of week 11 material. MLE. Asumptotic optimality.
Review.
Handout take-home portion of final.

Exam December 10, 7:30