Instructor: James Gentle

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

This course resumes where CSI 972 ends (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.

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

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 CSI972 inclass final (remember the username and password).

Maximum likelihood estimation.

94, 95, 96(a),96(g), 96(h), 97, 109, 118, 120, 152.

Comments, solutions(incomplete).

- The likelihood principle.
- Computational issues.
- EM.
- Problems with MLEs.

- Asymptotic properties of MLEs.

- The ECDF and some pointwise properties.
- Norms and metrics, with applications to CDFs.
- The DKW inequality and the Glivenko-Cantelli theorem.
- The ECDF as a MLE for the CDF.
- Statistical functionals and the bootstrap.

3, 5, 9, 17, 20, 21, 24, 27, 39, 59, 61, 63, 74, 86

Comments, solutions(incomplete).

Estimation in nonparametric models continued.

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

Statistical functionals, and estimation based on them.

Handout takehome portion of midterm,

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

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

Sample from a previous year.

**Reading assignment:** Read Shao, Sections 5.4 and 5.5.

Hypothesis testing.

Introduction.

1, 4, 5(a), 5(b), 12, 17, 27, 29, 37, 38, 51, 52(a), 58, 63, 69(a), 74, 93, 98, 107

Comments, solutions(incomplete).

UMP tests; Neyman-Pearson theory

UMP tests in exponential families.

Parametric tests; LR, Wald, score tests.

Nonparametric tests; asymptotic properties.

Bayesian approaches to hypothesis testing.

1, 2, 9, 18, 28, 29, 31, 40, 44, 48, 60, 63, 67, 79, 82, 93, 95, 101

Comments, solutions(incomplete).

Bootstrap methods.

Bayesian credible sets.

Handout takehome portion of final (due May 10).

Review various topics in estimation; GEE; variance estimation.

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

Sample from a previous year.