Math 405/607: Advanced Topics in Probability and Statistics

Time and Place:	13:00-13:50 MWF, Fenton 105
CRN:	13732 (Math 405), 15943 (Math 607)
Office:	215M Deady Hall
Office Hours:	9:00 - 9:50, 11:00 -11:50 MWF
Office Phone:	346-4701
Email:	shao@math.uoregon.edu
Web URL:	http://darkwing.uoregon.edu/~qmshao

Reference books:

• Billingsley, P. Probability and Measure , John Wiley & Sons, 1996
• Shao, J. Mathematical Statistics , Springer, 1999
• Serfling, R. Approximation Theorems of Mathematical Statistics . John Wiley & Sons, 1980
• Efron, B. An Introduction to the Bootstrap , Chapman & Hall, 1993

This course covers a broad range of fundamental results in probability theory that are useful in statistics, along with methods of proof and techniques of application. The manipulation of ``probability" theorems to obtain ``statistical" theorems is emphasized. It is hoped that, besides a knowledge of these basic probability and statistical theorems, an appreciation on the instrumental role of probability theory and a perspective on practical needs for its further development may be gained.

The first part of course focuses on foundations of probability theory and a variety of tools basic to statistics. Foremost are:

• The law of large numbers
• The central limit theorem
• Related results

The second part deals systematically with the usual statistics computed from a sample:

• The sample distribution function
• The sample moments
• The sample quantiles
• The order statistics
• U-statistics

The third part deals with some special topics in statistics, such as

• Bootstrap
• Monte Carlo method