Instructor: James Gentle

Appointments for individual consultations can be made by email to the instructor.

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.

**Recitations** (optional): Monday, 7:30pm, Research Building I, room 92

During the optional recitation periods students and/or the instructor will discuss exercises, especially those listed as "for practice and discussion". The instructor may also discuss some of the class notes.

If you send email to the instructor, please put "CSI 973" in the subject line.

This course is the second part of a two-course sequence. The general description of the two courses is available at mason.gmu.edu/~jgentle/csi9723/

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.

See also the references listed in the
general description.

This course resumes where CSI 972 ended (which is at the end of Section 4.3 in Shao).

The course begins with a brief review of the general theory of statistical estimation, and estimation in parametric models. It then continues with maximum likelihood estimation and asymptotic properties of estimators in parametric models. Next, estimation in nonparametric models is covered. Hypothesis tests and confidence intervals are then covered.

I have put together a set of notes to supplement the material in the text and the lectures. These notes have a subject index that should be useful. (I am continually working on these notes, so they may change from week to week.)

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

Each homework will be graded based on 100 points, and 5 points will be deducted for each day that the homework is late.

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.

Except during a period between when a take-home exam has been given out and when the exam is due, 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).
**
During a period between when a take-home exam has been given out and when the
exam is due, students are not to discuss with each other any aspect of the course
-- homework, examples, or anything else relating to the course in any way.
**
Any violation of this rule is a violation of the GMU Honor Code.

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.

Review principles and procedures of statistical estimation and other relevant material from CSI 972.

Asymptotic efficiency.

Maximum likelihood estimation.

Comments.

- The likelihood principle.
- Computational issues.
- Problems with MLEs.
- Variations on likelihood functions.

EM examples. (Read general notes on optimization.)

- MLE in generalized linear models.
- Quasi-likelihood and conditional likelihood.
- Asymptotic properties of MLEs.

The Bayesian approach; Bayesian estimation (review from MathStat_I, pp 77-95)

**Reading assignment:** Read Shao, Sections 4.1, 6.4.4, and 7.1.3.

**Assignment 2, due Mar 3: ** In Exercises 4.6: 31, 32 (note typo for (b) and (c));
in Exercises 6.6: 106, 107;
in Exercises 7.6: 28, 29.

May turn in as late as March 7.
Comments.

Followup comments on MLEs in parametric ranges.

Hand out midterm takehome.

Between now and the end of class on March 3, students are not to discuss homework or other aspects of the course (including the takehome of course!) with anyone other than the instructor.

Closed book and closed notes except for one sheet (front and back) of prewritten notes.

1, 5(a), 5(b), 5(c), 12, 21, 23, 27(a), 27(b), 27(c), 38, 52(a), 52(b), 63, 69(a), 74, 92(a), 92(b), 99(a), 99(b), 107

Hypothesis testing.

Because of St Patrick's Day, there will be no recitation session this evening.

UMP tests; Neyman-Pearson theory

UMP tests in exponential families.

Equivariance.

LR tests.

Asymptotic properties.

Wald and score tests.

1, 2, 28, 40, 44, 79, 82, 93, 95, 101

Comments.

3, 5, 9, 17, 20, 21, 24, 27, 39, 59, 61, 63, 74, 86 90, 96, 111 (note typo: \hat{c}_4 should be \hat{c}_4 - \hat{c}_2^2

Some useful statistical functionals: L, M, and R estimators.

Robust estimation.

Nonparametric density estimation.

- Miscellaneous topics
- Empirical likelihood and examples.
- Nonparametric probability density estimation.
- Partial likelihood and profile likelihood.

- Exercises

Closed book and closed notes except for one sheet of prewritten notes.