Theoretical Multivariate Statistics

(Master, 1st Semester, WS)

Most of the observable phenomena in empirical sciences are of a multivariate nature. For example, in financial studies, assets in the stock markets are simultaneously analyzed. In medicine, recorded observations of subjects in different locations are the basis of reliable diagnostics and medications. In quantitative marketing, consumer preferences are collected in order to construct models of consumer behavior. This course will teach you a collection of tools and procedures that involve observing and analyzing more than one statistical variable at a time.

We will treat the following topics: descriptive techniques, matrix algebra, regression analysis, simple analysis of variance, multivariate distributions, theory of the multinormal, theory of estimation, hypothesis testing, decomposition of data matrices by factors, principal components analysis. 

• Descriptive Statistics and Tests are important tools to make conclusions about the sample and the population. We will repeat known descriptive measures and tests first and introduce new ones later on. A case study will be presented.

• Regression analysis attempts to determine a linear formula that can describe how some variables respond to changes in others. Linear regression is a method for determining the parameters of a linear system. In this course, multiple linear regression is studied when there is more than one explanatory variable.

• Multivariate normal distribution, sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which is a generalization to higher dimensions of the one-dimensional normal distribution. The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem (the proof of which requires advanced undergraduate mathematics). Many psychological measurements and physical phenomena (like photon counts and noise) can be approximated well by the normal distribution. While the mechanisms underlying these phenomena are often unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation.

• The theory of estimation develops the basic theoretical tools needed to derive estimators and determine their properties in general situations. This part of the course mainly builds on the maximum likelihood theory. In many situations, the maximum likelihood estimator indeed shares asymptotic optimal properties making its use easy and appealing. The multivariate normal population and the linear regression model will be presented, where the applications are numerous and derivations are easy to do. In multivariate setups, the maximum likelihood estimator is, at times, too complicated to be derived analytically. In such cases, the estimators are obtained using numerical methods (nonlinear optimization). However, the general theory and the asymptotic properties of this estimator remain simple and valid.

• Hypothesis testing is used to test the hypothesis that the unknown parameter belongs to some subspace. A rejection region can be based on a likelihood ratio principle. Various testing problems and examples describe this technique .

Semester progress

• The course contains one lecture and one tutorial per week, self-study. 
• To get the credit points the participants have to attend the written exam (120 minutes).
• The course language is English.

OPAL

• The lecture script, lecture videos, and exercises are available in OPAL Kurs.
• Please register for the course in OPAL!

Schedule

Topics

• Matrix algebra.
• Moving to higher dimensions
  (regression analysis, simple analysis of variance)
• Multivariate distributions.
• Theory of multinormal.
• Copula.
• Theory of estimation.
• Hypothesis testing.

Literature

• Backhaus, K., Erichson, B., Plinke, W., Weiber, R. (2008),  Multivariate Analysis Methods: An Application-Oriented Introduction  (12th Edition), Springer Verlag.
• Hardle W, Simar L (2015)  Applied Multivariate Statistical Analysis  (4th edition) Springer textbook.
• Hardle W, Hlavka Z (2007)  Multivariate Statistics: Exercises and Solutions , Springer Textbook.