In mathematical statistics the aim is to analyze data sets (samples) to gain insight on a broader entirety. Students of this course will be introduced to the basics behind the theory of mathematical statistics in a comprehensive introduction. Essential estimation and test methods are presented. These methods will also be implemented in modern software languages accompanied by the application examples which will give students a deeper understanding of the theory. Furthermore, the goal is to teach all fundamentals needed for more advanced statistical purposes such as in biostatistics, actuarial sciences and finance.

Lecture

Lecturer
Prof. Dr. Evgeny Spodarev

Assistant
Dr. Michael Juhos

Time and Place

Lectures

Monday, 10–12, N24-H14,
Wednesday, 12–14, N24-H14.
First lecture takes place monday, October 14, from 10 am to 12 am in WWP 47.0.501 („Roter Hörsaal“).

Exercise session

Thursday, 10–12, N24-H12.

ECTS

4 hours lecture + 2 hours exercise session

Prerequisites

• Elementare Wahrscheinlichkeitsrechnung und Statistik
• Wahrscheinlichkeitstheorie und Stochastische Prozesse

Target Audience

• BSc Mathematik: Wahlpflicht Angewandte Mathematik
• BSc Wirtschaftsmathematik: Wahlpflicht Stochastik/Optimierung/Finanzmathematik
• BSc Mathematische Biometrie: Wahlpflicht Stochastik
• MSc Mathematik: Wahlpflicht Angewandte Mathematik
• MSc Wirtschaftsmathematik: Wahlplficht Stochastik/Optimierung/Finanzmathematik
• MSc Mathematische Biometrie: Wahlpflicht Mathematik und Statistik
• MSc Finance: Wahlpflicht Mathematik

The course Mathematical Statistics (lecture and exercises) takes place from 14/10/2024 all the semester long; in addition the first half of the course until 05/12/2024 (lecture and exercises) constitutes the Angewandte Stochastik II for the bachelor’s programme Computational Science and Engeneering and concludes with its own exam.

Contents

• Parametric models and fundamental theory
• Exponential families, completeness, sufficiency
• Point estimation
• Properties of estimators (MSE, bias, consistency)
• Best unbiased estimators, Cramer-Rao inequality
• U-statistics, confidence intervals
• Hypothesis testing
• Density estimation or linear models (introduction)

Lecture Notes

German version

English version

Exercise Sheets

See Moodle website.

Exam

To participate in the final exam at least 50% of the homework points need to be achieved. Further information on Moodle website.

Registration for the exam needs to be completed four days in advance of the exam date (only possible if the prerequisite for the exam is passed).

Exam dates:

Literature

• H. Dehling and B. Haupt, 
  Einführung in die Wahrscheinlichkeitstheorie und Statistik
  Springer, Berlin, 2003.
   
• P. Bickel and K. Doksum
  Mathematical Statistics: Basic Ideas and Selected Topics 
  Prentice Hall, London, 2001. 2nd ed., Vol. l.
   
• A. A. Borovkov,
  Mathematical Statistics 
  Gordon & Breach, 1998.
   
• G. Casella and R. L. Berger,
  Statistical Inference 
  Pacific Grove (CA), Duxbury, 2002.
   
• E. Cramer and U. Kamps, 
  Grundlagen der Wahrscheinlichkeitsrechnung und Statistik 
  Springer, Berlin, 2007.
   
• P. Dalgaard, 
  Introductory Statistics with R 
  Springer, Berlin, 2002.
   
• A. J. Dobson, 
  An Introduction to Generalizes Linear Models 
  Chapmen& Hall, Boca Raton, 2002.
   
• L. Fahrmeir and T. Kneib and S. Lang,
  Regression. Modelle, Methoden und Anwendungen 
  Springer, Berlin, 2007.
   
• L. Fahrmeir and R. Künstler and I. Pigeot and G. Tutz,
  Statistik. Der Weg zur Datenanalyse 
  Springer, Berlin, 2001.
   
• H. O. Georgii, 
  Stochastik 
  de Gruyter, Berlin, 2002.
   
• J. Hartung and B. Elpert and K. H. Klösener, 
  Statistik. R 
  Oldenbourg Verlag, München, 1993. 9. Auflage.
   
• C. C. Heyde and 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 and J. Braun,
  Data Analysis and Graphics Using R 
  Cambridge University Press, 2003.
   
• M. Overbeck-Larisch and W. Dolejsky, 
  Stochastik mit Mathematica 
  Vieweg, Braunschweig, 1998.
   
• H. Pruscha,
  Angewandte Methoden der Mathematischen Statistik 
  Teubner, Stuttgart, 2000.
   
• H. Pruscha, 
  Vorlesungen über Mathematische Statistik 
  Teubner, Stuttgart, 2000.
   
• L. Sachs, 
  Angewandte Statistik 
  Springer, 2004.
   
• L. Sachs and J. Hedderich, 
  Angewandte Statistik, Methodensammlung mit R
  Springer, Berlin, 2006.
   
• Robert J Serfling, 
  Approximation theorems of mathematical statistics 
  volume 162. John Wiley & Sons, 2009.
   
• M. R. Spiegel and L. J. Stephens, 
  Statistik 
  McGraw-Hill, 1999.
   
• Spokoiny and Dickhaus, 
  Basics of modern mathematical statistics 
  Springer, 2015.
   
• W. A. Stahel, 
  Statistische Datenanalyse
  Vieweg, 1999.
   
• W. Venables and D. Ripley, 
  Modern applied statistics with S-PLUS 
  Springer, 1999. 3rd ed.
   
• L. Wasserman, 
  All of Statistics. A Concise Course in Statistical Inference 
  Springer, 2004.

Other useful literature to this course can be found in the Semesterapparat.