Concepts and treatment of random vectors

Teacher(s)

von Sachs Rainer;

Language

French

Prerequisites

Concepts and tools equivalent to those taught in the UE LSTAT2014: Elements of probability and mathematical statistics

Main themes

Concepts of random vectors, multivariates moments and distributions, dependecies - preparing the student for the concept of dependence (prerequisite for many courses of the Master in Statistics)

Learning outcomes

At the end of this learning unit, the student is able to :

determine the joint, marginal and conditional distributions, as well as their associated moments.

understand the concepts of dependence quantified by the (partial) correlation and compare them to the case of independence.

master classical multivariate distributions such as multinormal and multinomial.

perform theoretical calculations by hand and using statistical software.

Content

Joint probability distributions: discrete, continuous
Marginal distributions, conditional distributions
Independence
Covariance and correlation
Moments (moment generating functions) 
Conditional moments (expectation and variance)
Functions of random vectors, transformations
 Multinomial distribution
Multivariate normal distribution: construction, properties
Theory of multinormal: conditional normal, partial correlation, precision matrix, conditional independence
Other dependence concepts: copulas

Teaching methods

The material will be treated from a theoretical point of view, but also via examples and exercices (including simulations on R).

Evaluation methods

The exam will consist in a written exam, completed by a projet on simulations (with R).

Online resources

Moodle (copies of slides, ...)

Bibliography

Chapitres 4.1-4.4 et 4.7 , 5.1- 5.2 (5.3-5.4) du livre « Applied Multivariate Statistical Analysis » (W. Härdle, L. Simar ; Springer 2007) ;
Chapitres 2.5-2.8, 3.5-3.6, 3.9-3.11 ; 4.1.4 et 4.3 du livre « Mathematical Statistics for Economics and Business” (R. Mittelhammer ; Springer 2013)

Teaching materials

• copies des slides sur Moodle

Faculty or entity

LSBA