Multivariate Models


Acquire basic knowledge in multivariate statistics, particularly about the multivariate Normal and Wishart distributions, inference about mean vectors, multivariate regression, canonical analysis and introduction to copulas.
Use the concepts and techniques properly apprehended, along with the R software, in solving most diverse problems of real life.

General characterization





Responsible teacher

Filipe José Gonçalves Pereira Marques


Weekly - 4

Total - 56

Teaching language



Good knowledge of the subjects taught in Linear Algebra and in an introductory Course on Probabilities and Statistics.


Johnson, R. and Wichern, D. W. (2007), Applied Multivariate Statistical Analysis, 6th Edition, Prentice Hall, New Jersey.

Flury, B. (1997), A First Course in Multivariate Statistics, Springer. New York

Morrison, D. F. (2004), Multivariate Statistical Methods, 4th Edition, Duxbury Press

Rencher, A. C. (1998), Multivariate Statistical Inference and Applications, John Wiley & Sons

Rencher, A. C. (2002), Methods of Multivariate Analysis, John Wiley & Sons

Teaching method

Lectures, where the main concepts and results will be introduced to the students, together with the presentation of illustrative examples, which are intended to enlighten the concepts presented. These Lectures will be complemented with the resolution of problems with the help of an adequate computer software, related to the concepts introduced in the Lectures and will be made with the active participation of the students.

Evaluation method


1. Pre-Requisites

In order to be able to have access to the course evaluation, both to midterms and tests and also to the Exam, students need to have the presence in at least 2/3 on the presential classes.


 2. Evaluation 

The recommended form of evaluation consists in:

1º Test - weight: 50%

2º Test - weight: 50%

The student who has an average grade (weighted mean) of at least 9,5 (on a 0-20 scale) will be approved in the course.Students who obtained a final grade from tests less than 9.5 (on a 0-20 scale), may have access to a final Exam, in case they have attended at least 2/3 of Labs and 2/3 of Classes.Also the students who had a grade equal or greater than 9.5 from tests may have access to the Final Exam in order to improve their grade. Students with a final grade of more than 18 (on a 0-20 scale) have to go through an oral examination, or their final grade will be equal to 18.


Subject matter

1 - Brief reviews and basics of Linear Algebra
2 - The Multivariate Normal distribution. Maximum likelihood estimators and their distributions. The Wishart distribution
3 - Inference on vectors of averages
3.1 - Tests based on a sample
3.2 - Tests based on two samples, paired samples and independent samples
3.3 - Tests based on multiple samples
4 - Multivariate Regression and Canonical Analysis
5 - Introduction to Copulas
5.1 - Definition
5.2 - Dependence, Concordance, Upper and Lower tail dependencies
5.3 - The Guassian Copula
5.4 - The Archimedean family of copulas