Description

Basic theories of statistical estimation, including optimal estimation in finite samples and asymptotically optimal estimation. A careful mathematical treatment of the primary techniques of estimation utilized by statisticians.

Prerequisites

Math 4261 and Math 4262 or equivalent

Topic Outline

The topics to be covered are as follows:

Statistical decision theory: geometry of decision problems, the fundamental theorem of game theory and its use in statistical decision theory, specialized techniques for finding minimax and Bayes estimators in standard problems of estimation
The Bayesian viewpoint: solving the no-data problem and using it in univariate and multivariate settings, detailed analysis for conjugate priors
Optimality under restrictions:
- Minimum variance unbiased estimation: the Rao-Blackwell and Lehmann-Scheffe theorems
- Equivariant estimation: invariance of statistical problems under groups and some applications in estimation
Asymptotic theory of estimation:
- General notions of asymptotic optimality: Hodges counterexample
- Le-Cam's theorem on asymptotic optimality
- Asymptotic optimality of maximum likelihood estimators, special cases including logistic regression
- Robust estimators (M, L, and R) and their asymptotic relative efficiencies
- Asymptotic optimality of Bayes estimators including higher order analysis characterizing asymptotic posterior distributions

Grading

Your grade for the course will be determined by your performance on two midterm exams, and a final exam. Each will count one third. Problems will be assigned during the semester both from the course text, Mathematical Statistics, Vol. 1, second edition, Prentice Hall, 2001, by Peter Bickel and Kjell Doksum, and other sources. Although problems will not be graded, selected solutions will be posted. A good way to study for the exam is to listen to the lectures, work the examples from class and the notes, and work the problems.

Some references are: Berger, J.W. (1985). Statistical Decision Theory and Bayesian Analysis. Springer- Verlag,New York.

Blackwell, D. and Girshick, M. (1954). Theory of Games and Statistical Decisions. Wiley,New York.

DeGroot, M. (1970) Optimal Statistical Decisions. McGraw-Hill, New York.

Ferguson, T. S. (1967) Mathematical Statistics, a Decision Theoretic Approach. Academic Press, New York. Lehmann, E.L. (1959). Testing Statistical Hypotheses. Wiley,New York.

Lehmann, E.L. (1983). Theory of Point Estimation. Wiley,New York.

Lehmann, E.L. and Casella, G. Theory of Point Estimation,