Mathematical Statistics

(= Stochastics III)

Whereas measure theory and probability theory (see the lectures "Stochastics" and "Stochastics II") laid the basis for modelling random processes and evolutions, there remains the problem of adapting those models to observed data by suitable statistical methods.Some basic concepts, methods and elementary theorems of statistics have been presented in "Stochastics II" in the context of parameter estimation and hypothesis testing.

The lecture "Mathematical Statistics" considers statistical methods from a more profound theoretical point of view and investigates, in particular, the problem of optimality: characterizi1ng th1e optimality of statistical methods, describing suitable assumptions for optimality, deriving optimum statistical procedures for general or special situations etc. This investigation is performed in the framework of statistical decision theory which is presented in a mathematical style, but with appropriate examples from practice. Another major point relates to the asymptotic theory, i.e., the performance of statistical methods for an increasing number of samples (asymptotic theory) and their behaviour in non-parametric or non-standard situations.

The main topics are characterized by the following chapter headings:

Statistical decision theory: loss and risk, optimality criteria, Bayesian procedures, minimax decision rules, multiple decision problems (classification)
Statistical testing procedures: Neyman-Pearson theory, linear optimization
Maximum likelihood theory: asymptotics, optimum estimators and tests
Linear models in statistics
Conditional expectation and conditional distributions
Optimality using sufficiency and completeness
Non-parametric and robust statistical methods

Literature:

FERGUSON, T.S.: Mathematical statistics. A decision-theoretical approach. Academic Press, New York, 1967
LEHMANN, E.L.: Theory of point estimation. Wiley, New york, 1983.
PRUSCHA, H.: Angewandte Methoden der mathematischen Statistik. Teubner, Stuttgart, 1989.
SERFLING,R.j.: Approximation theorems of mathematical statistics. Wiley, New York, 1980, 371 pp.
SCHERVISH, M.J.: Theory of statistics. Springer-Verlag, Berlin, 1995, 702 pp.
WINKLER,W.: Vorlesungen der mathematischen Statistik. Teubner, Stuttgart, 1983.
WITTING, H.: Mathematische Statistik I. Teubner, Stuttgart, 1985, 538 S.

Lecture designed for:
Students of mathematics (4th to 8th semester), of sciences, of engineering and of Operations Research. Some preknowledge in probability and basic statistics (as presented in the previous lectures Stochastics I and II) is assumed. A list of contents is available at the institute's office.

Exercises:
Exercise sheets will be issued every week for being worked out at home, correction and discussion one week later. Successful candidates will get a certification provided that they participate regularly in the discussion groups and give an oral presentation of exercises.

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