# Statistical Theory – MATH 442

#### Instructor: Dr. Erwan Koch

#### Assistant: Tomáš Rubín

#### Description

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.

#### Topics include:

- 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.

#### Learning outcomes

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.

#### Teaching methods

Lecture ex cathedra using slides as well as the blackboard (especially for some proofs). Examples/exercises presented/solved at the blackboard.

#### Assessment methods

Final written exam.

#### Recommended Texts

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 |
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Exercises: | MA A1 12 | Tuesdays, 10:15-12:00 |