Theoretical Multivariate Statistics (WS)
Tasks of Multivariate Statistics
Most of the observable phenomena in the empirical sciences are of a multivariate nature. In financial studies, assets in the stock markets are observed simultaneously and their joint development is analyzed to better understand general tendencies and to track indices. In medicine recorded observations of subjects in different locations are the basis of reliable diagnoses and medication. In quantitative marketing consumer preferences are collected in order to construct models of consumer behavior. The underlying theoretical structure of these and many other quantitative studies of applied sciences is multivariate. The course of Theoretical Multivariate Analysis describes a collection of procedures which involve observation and analysis of more than one statistical variable at a time.
In this course following topics are treated: 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. Descriptive measures and known test will be repeated and new descriptive measures and tests will be introduced. 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 a case of multiple linear regression is studied, when there are more than one explanatory variable.
 Multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which is a generalization to higher dimensions of the onedimensional 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
 Theory of estimation develops the basic theoretical tools which are needed to derive estimators and to determine their properties in general situations. This part of course basically relies on the maximum likelihood theory. In many situations, the maximum likelihood estimators indeed share asymptotic optimal properties which make their 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). The general theory and the asymptotic properties of these estimators remain simple and valid.
 Hypothesis testing combines statistical tools which allow to test the hypothesis that the unknown parameter belongs to some subspace. A rejection region can be constructed based on likelihood ratio principle. This technique will be illustrated through various testing problems and examples such as comparison of several means, repeated measurements and profile analysis.
News
 Please register for the course in OPAL.
 For questions please contact Iryna Okhrin.
Schedule
Event  Day  Time  Room  Teacher 

Lecture 
until Friday 
 
online 

Excersise 
until Friday 
 
online 
Themen
 Matrix algebra.
 Moving to higher dimensions.
 Linear Models and ANOVA.
 Multivariate Distributions.
 Theory of Multinormal.
 Multivariate Distributions.
 Copula.
 Theory of Estimation and EM Algorithm.
 Hypothesis Testing.
Literatur

Backhaus, K., Erichson, B., Plinke, W., Weiber, R. (2008), Multivariate Analysemethoden: Eine anwendungsorientierte Einführung (12. Auflage), Springer Verlag.

Härdle, W., Simar, L. (2015), Applied Multivariate Statistical Analysis (4nd edititon), Springer Lehrbuch.

Härdle, W., Hlavka, Z. (2007), Multivariate Statistics: Exercises and Solutions, Springer Lehrbuch.