Advanced Topics in Multivariate Analysis
Code
9706
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Filipe José Gonçalves Pereira Marques
Weekly hours
4
Total hours
56
Teaching language
Português
Objectives
To give students a good perspective of what are the likelihood ratio tests in multivariate analysis, as well as the ability to test for elaborate structures and hypotheses, through the decomposition of the hypotheses and corresponding tests into adequate sequences of hypotheses and tests for which we may know the corresponding likelihood ratio statistics and through the composition of which they will be able to obtain both the overall likelihood ratio test statistic and very good approximations for their distributions, given that their exact distributions are by far non-manageable.
Prerequisites
Knowledge of Probability and Statistics at the level of those obtained at the courses Probability and -statistics I and II of the undergraduate degree in Mathematics.
Subject matter
- Part I -- Brief overview of the ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''basic'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' multivariate distributions
- the multivariate Normal distribution
- the Maximum Likelihood Estimators of the parameters
- the Wishart distribution
- (Bibliography: Kshirsagar (Cap. 2,3); Muirhead (Cap. 1,3); Anderson (Cap. 2,3); small volume of class notes)
- the multivariate Normal distribution
- Part II -- Likelihood ratio tests and distribution of the corresponding test statistics (exact, asymptotic and near-exact distributions)
- the ''''''''''''''''basic'''''''''''''''' likelihood ratio tests
- the test of independence of several groups of variables
- the test of equality of several mean vectors
- the test of equality of several covariance matrices
- the sphericity test
- (Bibliography: Kshirsagar (Cap. 8,10); Muirhead (Cap. 8); Anderson (Cap. 9,10); small volume of class notes)
- Composition of tests and hypotheses
- advantages of the approach
- the sphericity test revisited
- the test of equality of several multivariate Normal distributions
- families of tests and tests for elaborate covariance structures (and their particular cases)
- the multisample block-matrix sphericity test
- the multisample block-scalar sphericity test
- the multisample hyperblock-matrix sphericity test
- (Bibliography: papers)
- the ''''''''''''''''basic'''''''''''''''' likelihood ratio tests
Bibliography
- Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed., Wiley Interscience, New York.
- Kshirsagar, A. M. (1972). Multivariate Analysis, Marcel Dekker, New York.
- Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, J. Wiley & Sons, New York.
- Coelho, C. A. (2001). Tópicos em Estatística Multivariada e Métodos Estatísticos de Análise Multivariada (class notes - in Portuguese).
Teaching method
The course will be taught through mixed lecture-lab classes where together with the exposition of the main concepts and results the main likelihood ratio tests used in Multivariate Analysis will be obtained as well as also other tests not usually available in the literature, all this followed by illustrative examples. These examples will then be used as the starting point for sets of problems, some of which are intended to be solved in class, with the active participation of the students, while others are intended to be left as challenges, which the students are supposed to solve by recurring to some papers suggested as supplementary bibliography. It is intended that this later set of problems will change from year to year.
Evaluation method
Two sets of problems to be solved individually, with a weight of 30% each for the final grade, together with the writing of a report, to be written individually by each student, on a topic given by the Professor, which will have the weight of 40% for the final grade. The report will be written in English, with the objective to be submitted to an international journal in the area of Statistics. This journal has to one that is indexed in Mathematical Reviews/MathSciNet.