Statistical Theory – MATH 442
Instructor: Dr. Erwan Koch
Assistant: Tomáš Rubín
The course aims at developing certain key aspects of the theory of statistics, providing a common general framework for statistical methodology. While the main emphasis will be on the mathematical aspects of statistics, an effort will be made to balance rigor and intuition.
- Stochastic convergence and its use in statistics: modes of convergence, weak law of large numbers, central limit theorem.
- Formalization of a statistical problem : parameters, models, parametrizations, sufficiency, ancillarity, completeness.
- Point estimation: methods of estimation, bias, variance, relative efficiency.
- Likelihood theory: the likelihood principle, asymptotic properties, misspecification of models, the Bayesian perspective.
- Optimality: decision theory, minimum variance unbiased estimation, Cramér-Rao lower bound, efficiency, robustness.
- Testing and Confidence Regions: Neyman-Pearson setup, likelihood ratio tests, uniformly most powerful (UMP) tests, duality with confidence intervals, confidence regions, large sample theory, goodness-of-fit testing.
Required prior knowledge
Real Analysis, Linear Algebra, Probability, Statistics.
By the end of the course, the student must be able to:
- Formulate the various elements of a statistical problem rigorously.
- Formalize the performance of statistical procedures through probability theory.
- Systematize broad classes of probability models and their structural relation to inference.
- Construct efficient statistical procedures for point/interval estimation and testing in classical contexts.
- Derive certain exact (finite sample) properties of fundamental statistical procedures.
- Derive certain asymptotic (large sample) properties of fundamental statistical procedures.
- Formulate fundamental limitations and uncertainty principles of statistical theory.
- Prove certain fundamental structural and optimality theorems of statistics.
Lecture ex cathedra using slides as well as the blackboard (especially for some proofs). Examples/exercises presented/solved at the blackboard.
Final written exam.
Bickel. P.J. & Doksum, K.A. (2000). Mathematical Statistics: Basic Ideas and Selected Topics, Volume I. Prentice Hall.
Knight, K. (2000). Mathematical Statistics. Chapman and Hall.
Cox, D.R. & Hinkley, D.V. (1979). Theoretical Statistics. Chapman & Hall.
Fall 2019 Schedule
|Lectures:||MA A1 12||Tuesdays, 8:15-10:00|
|Exercises:||MA A1 12||Tuesdays, 10:15-12:00|