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Selected Topics of Multivariate Analysis

Objectives

Give the students an introduction to some of the most important inferential techniques and models used in Multivariate Analysis through the development of geometrical and algebraic and mostly inferential approaches to such models. The students are supposed to acquire:
• a global understanding of multivariate generalizations of univariate tests and models, through the study of the Canonical Analysis or Multivariate Regression and Multivariate Variance and Covariance models;
• knowledge of the inferential techniques and tests of fit and tests to parameters in such models;
• knowledge of the Wilks Lambda statistic and its distribution, namely in situations of high dimensionality where p/n tens to 1;
• knowledge of techniques and models to be applied in high dimensionality and big data situations, as well as dimensionality reduction techniques (eg: Factorial Analysis and PCA);
• familiarization with softwares to implement the tests and models introduced.

General characterization

Code

13000

Credits

9.0

Responsible teacher

Carlos Manuel Agra Coelho

Hours

Weekly - 4

Total - 14

Teaching language

Português

Prerequisites

A median knowledge of Univariate Statistics, namely that of the techniques related with the derivation of expressions for expected values and moments of random variables, as well as those related with the derivation and handling of moment generating functions, as well as yet the techniques related with the transformation of random variables.

At least a basic knowledge of Mathematical Analysis and Linear Algebra are also required.

Bibliography

1. Anderson, T.W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. J. Wiley & Sons
2. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. J. Wiley & Sons
3. Marques, F.J., Coelho, C.A., Arnold, B.C. (2011). A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis, TEST,20,180-203
4. Srivastava, M.S. (2007). Multivariate theory for analyzing high dimensional data. J.Jap.Stat. Soc.,37,53-86
5. Niu, Z., Hu, J., Bai, Z., Gao, W. (2019). On LR simultaneous test of high-dimensional mean vector and covariance matrix under non-normality. Stat.Prob.Lett.,145,338-344.
6. Rauf, A.M. (2019). Multiple comparisons of mean vectors with large dimension under general conditions. J.Stat.Comp.Simul.,89,1044-1059
7. Zhong, P.-S., Lan, W., Song, P.X.K., Tsai, C.-L. (2017). Tests for covariance structures with high-dimensional repeated measurements. Ann. Statist. 45,1185-1213.

Teaching method

All classes, even those of a more theoretical type, will always be accompanied by examples and exercises that allow the students to get a better understanding of the subjects being taught, being also proposed in each class take home problems for the students to solve, which will also be delineated in such a way that may give the students a tool to deepen their understanding and knowledge of the techniques and models taught in class. Indeed it is incentivized the collaborative resolution of at least some of these problems, being anyway its resolution part of the evaluation of the course. 

Evaluation method

 In all classes will be proposed take home problems for the students to solve, which will also be delineated in such a way that may give the students a tool to deepen their understanding and knowledge of the techniques and models taught in class. Indeed it is incentivized the collaborative resolution of at least some of these problems, being its resolution part of the evaluation of the course, with a weight of 10%. Besides these problems, 2 or 3 other more elaborate problems will be proposed to students, which resolution should be done on an individual basis and which will also be part of the evaluation component of the course, with a weight of 30%. The evaluation component will also count with the realization of a test, with a weight of 50%, and the public presentation of the resolution of one of the more elaborate problems proposed, which will have a weight of 10%.

Subject matter

1. The Multivariate Normal and Wishart distributions. Some properties
2. Maximum Likelihood Estimators of the parameters of a Multivariate Normal distribution and their distributions
3. The Hotelling T2 statistic. Applications: tests on vectors of expected values
4. The Generalized Canonical Analysis as an all-embracing multivariate linear model
a. The Wilks Lambda statistic and its exact, asymptotic and near-exact distributions and its distribution in cases where p/n tends to 1
b. Other models as particular cases (Multivariate Analysis of Variance and Covariance and Discriminant Analysis)
5. Tests for equality of mean vectors and tests for covariance structures for high dimensionality
6. The Dempster test for p > n and the tests of Fujikoshi et al. (2004), Schott (2007), Chen and Qin (2010) and Zhang et al. (2020).
7. Dimensionality reduction techniques. The Factorial Analysis and Principal Components Analysis