Distribuições Laplace assimétricas deslocadas matriciais: desenvolvimento, propriedades e modelos de mistura finita

Abstract

This thesis presents a new multivariate distribution, the multivariate shifted generalized asymmetric Laplace (SGAL) distribution, and two new matrix-variate distributions for modeling complex data exhibiting skewness and heavy tails: the matrix-variate shifted asymmetric Laplace (MVSAL) distribution and the matrix-variate shifted generalized asymmetric Laplace (MVSGAL) distribution. The proposed distributions belong to the class of normal mean-variance mixtures, resulting in a hierarchical representation that facilitates random data generation and likelihood-based inference. Several structural properties are established, including closure under linear transformations and marginalization. For the SGAL distribution, we present a finite mixture model estimated via the EM algorithm. The results, both from the simulation study and the application to real data from the Old Faithful geyser eruption, demonstrate good recovery of the location and skewness parameters, but indicate instability in the estimation of the shape and scale parameters. These results reinforce the need for more stable reparameterizations, such as fixing certain mixture parameters, in order to reduce model complexity and increase the robustness of the estimation process. This motivation led directly to the formulation of the reparameterized MVSGAL distribution for application in a finite mixture model. Within the inferential framework, ECM algorithms were developed for both the MVGAL distribution and the finite mixture model of the reparameterized MVGAL (MVSGAL-Ms). The simulation studies conducted throughout the thesis showed that the proposed algorithms are capable of adequately recovering the parameters generating the models. It was consistently observed that the quality of the estimates improves as the sample size increases. In addition, the studies showed that the MVSGAL distribution and the MVSGAL-Ms model offer high flexibility for representing data with different patterns of dependence, skewness, heterogeneity, and heavy tails. The empirical applications demonstrated the practical potential of the proposed MVSGAL distribution and MVGAL-Ms model. In the analysis of climatic data from municipalities in the state of São Paulo, they were able to identify seasonal patterns and distinct regimes, revealing interpretable structures of temporal dependence and associations among climatic variables.