Statistics deals with the question of how information about a larger whole can be obtained from data sets (samples) using mathematical methods. In this module students should learn about, understand, and apply the fundamentals of the theory of mathematical statistics and become familiar with the most important estimation and test procedures. They should be able to apply the methods in practice, especially with modern software. Furthermore the basis for advanced statistical considerations (especially of a bio- and econometric kind) should be comprehensively learnt and references to other mathematical fields should be recognised and used.

Organiser

Lecturer
Prof Dr Evgeny Spodarev

Assistant
Dr Michael Juhos

Time and place

Lecture
Monday, 10–12, N24 H14
Thursday, 10–12, N24 H12

Exercise
Wednesday, 12–14, N24 H14

Length

Four hours of lecture and two hours of exercises per week.

Prerequisites

• Elementary probability and statistics,
• Probability theory and stochastic processes.

Target audience

Mathematical Statistics (lecture and exercise) runs the whole semester from 13/10/2025; in addition, the first half of the course until 04/12/2025 (lecture and exercise) forms Applied Stochastics II for the bachelor’s programme Computational Science and Engineering and concludes with its own exam.

Contents

• Parametric model and its fundamentals
• Exponential families, completeness, sufficiency
• Methods for (point) estimation of parameters
• Quality properties of estimators (MSE, bias, consistency, ...)
• Best unbiassed estimator, Cramer–Rao inequality
• Tests of statistical hypotheses, relationship between tests and confidence intervals
• Density estimation or linear models (introductory)

Lecture notes

German version

English version

Exercise sheets

Exercise sheets and further information will be uploaded on the Moodle website.

Written exam

The condition for participation in either exam is passing the prerequisites. For that at least 50% of the exercise points must be achieved.

Exam dates (provisional):

Literature

• H. Dehling, B. Haupt,
  Introduction to Probability Theory and Statistics
  Springer, Berlin, 2003.
   
• P. Bickel, K. Doksum,
  Mathematical Statistics: Basic Ideas and Selected Topics
  Volume 1. Prentice Hall, London,^2nd edition 2001.
   
• A. A. Borovkov,
  Mathematical Statistics
  Gordon & Breach, 1998.
   
• G. Casella, R. L. Berger,
  Statistical Inference
  Pacific Grove (CA), Duxbury, 2002.
   
• E. Cramer, U. Kamps,
  Fundamentals of Probability and Statistics
  Springer, Berlin, 2007.
   
• P. Dalgaard,
  Introductory Statistics with R
  Springer, Berlin, 2002.
   
• A. J. Dobson,
  An Introduction to Generalised Linear Models
  Chapmen& Hall, Boca Raton, 2002.
   
• L. Fahrmeir, T. Kneib, S. Lang,
  Regression. Models, Methods and Applications
  Springer, Berlin, 2007.
   
• L. Fahrmeir, R. Künstler, I. Pigeot, G. Tutz.
  Statistics. The path to data analysis
  Springer, Berlin, 2001.
   
• H. O. Georgii,
  Stochastics
  de Gruyter, Berlin, 2002.
   
• J. Hartung, B. Elpert, K. H. Klösener,
  Statistik. R
  Oldenbourg Verlag, Munich, 9th condition 1993.
   
• C. C. Heyde, E. Seneta,
  Statisticians of the Centuries
  Springer, Berlin, 2001.
   
• A. Irle,
  Wahrscheinlichkeitstheorie und Statistik, Grundlagen, Resultate, Anwendungen
  Teubner, 2001.
   
• I. T. Jolliffe,
  Principal component analysis
  Springer,^2nd edition 2002.
   
• K. R. Koch,
  Parameter Estimation and Hypothesis Testing in Linear Models
  Springer, Berlin, 1999.
   
• E. L. Lehmann,
  Elements of Large-Sample Theory
  Springer, New York, 1999.
   
• J. Maindonald, J. Braun,
  Data Analysis and Graphics Using R
  Cambridge University Press, 2003.
   
• M. Overbeck-Larisch, W. Dolejsky,
  Stochastics with Mathematica
  Vieweg, Braunschweig, 1998.
   
• H. Pruscha,
  Angewandte Methoden der Mathematischen Statistik
  Teubner, Stuttgart, 2000.
   
• H. Pruscha,
  Lectures on Mathematical Statistics
  Teubner, Stuttgart, 2000.
   
• L. Sachs,
  Applied Statistics
  Springer, 2004.
   
• L. Sachs, J. Hedderich,
  Angewandte Statistik, Methodensammlung mit R
  Springer, Berlin, 2006.
   
• R. J. Serfling,
  Approximation theorems of mathematical statistics
  Volume 162. John Wiley & Sons, 2009.
   
• M. R. Spiegel, L. J. Stephens
  Statistics
  McGraw-Hill, 1999.
   
• V. Spokoiny, T. Dickhaus,
  Basics of modern mathematical statistics
  Springer, 2015.
   
• W. A. Stahel,
  Statistical Data Analysis
  Vieweg, 1999.
   
• W. Venables, D. Ripley
  Modern applied statistics with S-PLUS
  Springer,^3rd edition 1999.
   
• L. Wasserman,
  All of Statistics. A Concise Course in Statistical Inference
  Springer, 2004.

Further literature suggestions in the course reserves (Semesterapparate).