Goodness-of-fit and correlation tests are considered for dependent univariate data that arises when multivariate data is projected to the real line with a data-suggested linear transformation. Specifically, tests for multivariate normality are investigated. Let { } i Y be a sequence of independent k-variate normal random vectors, and let 0 d be a fixed linear transform from Rk to R . For a sequence of linear transforms { ( )} 1 , , n d Y Y converging almost surely to 0 d , the weak convergence of the empirical process of the standardized projections from d to a tight Gaussian process is established. This tight Gaussian process is identical to that which arises in the univariate case where the mean and standard deviation are estimated by the sample mean and sample standard deviation (Wood, 1975). The tight Gaussian process determines the limiting null distribution of E.D.F. goodness-of-fit statistics applied to the process of the projections. A class of tests for multivariate normality, which are based on the Shapiro-Wilk statistic and the related correlation statistics applied to the dependent univariate data that arises with a data-suggested linear transformation, is also considered. The asymptotic properties for these statistics are established. In both cases, the statistics based on random linear transformations are shown to be asymptotically equivalent to the statistics using the fixed linear transformation. The statistics based on the fixed linear transformation have same critical points as the corresponding tests of univariate normality; this allows an easy implementation of these tests for multivariate normality. Of particular interest are two classes of transforms that have been previously considered for testing multivariate normality and are special cases of the projections considered here. The first transformation, originally considered by Wood (1981), is based on a symmetric decomposition of the inverse sample covariance matrix. The asymptotic properties of these transformed empirical processes were fully developed using classical results. The second class of transforms is the principal components that arise in principal component analysis. Peterson and Stromberg (1998) suggested using these transforms with the univariate Shapiro-Wilk statistic. Using these suggested projections, the limiting distribution of the E.D.F. goodness-of-fit and correlation statistics are developed.

Submitted by Rafael Santana (rafael.silvasantana@ufpe.br) on 2017-04-26T17:07:37Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) / Made available in DSpace on 2017-04-26T17:07:38Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) Previous issue date: 2016-10-31 / CAPES; BOLSA DO CHILE. / In the last decades, univariate Birnbaum-Saunders models have received considerable attention in the literature. These models have been widely studied and applied to fatigue, but they have also been applied to other areas of the knowledge. In such areas, it is often necessary to model several variables simultaneously. If these variables are correlated, individual analyses for each variable can lead to erroneous results. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. In addition, diagnostic analysis is an important aspect to be considered in the statistical modeling. Furthermore, multivariate quality control charts are powerful and simple visual tools to determine whether a multivariate process is in control or out of control. A multivariate control chart shows how several variables jointly affect a process. First, we propose, derive and characterize multivariate generalized logarithmic Birnbaum-Saunders distributions. Also, we propose new multivariate generalized Birnbaum-Saunders regression models. We use the method of maximum likelihood estimation to estimate their parameters through the expectation-maximization algorithm. We carry out a simulation study to evaluate the performance of the corresponding estimators based on the Monte Carlo method. We validate the proposed models with a regression analysis of real-world multivariate fatigue data. Second, we conduct a diagnostic analysis for multivariate generalized Birnbaum-Saunders regression models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. Moreover, we consider the local influence method and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in the R software, which are illustrated with real-world multivariate biomaterials data. Third and finally, we develop a robust methodology based on multivariate quality control charts for generalized Birnbaum-Saunders distributions with the Hotelling statistic. We use the parametric bootstrap method to obtain the distribution of this statistic. A Monte Carlo simulation study is conducted to evaluate the proposed methodology, which reports its performance to provide earlier alerts of out-of-control conditions. An illustration with air quality real-world data of Santiago-Chile is provided. This illustration shows that the proposed methodology can be useful for alerting episodes of extreme air pollution. / Nas últimas décadas, o modelo Birnbaum-Saunders univariado recebeu considerável atenção na literatura. Esse modelo tem sido amplamente estudado e aplicado inicialmente à modelagem de fadiga de materiais. Com o passar dos anos surgiram trabalhos com aplicações em outras áreas do conhecimento. Em muitas das aplicações é necessário modelar diversas variáveis simultaneamente incorporando a correlação entre elas. Os modelos de regressão multivariados são uma ferramenta útil de análise multivariada, que leva em conta a correlação entre as variáveis de resposta. A análise de diagnóstico é um aspecto importante a ser considerado no modelo estatístico e verifica as suposições adotadas como também sua sensibilidade. Além disso, os gráficos de controle de qualidade multivariados são ferramentas visuais eficientes e simples para determinar se um processo multivariado está ou não fora de controle. Este gráfico mostra como diversas variáveis afetam conjuntamente um processo. Primeiro, propomos, derivamos e caracterizamos as distribuições Birnbaum-Saunders generalizadas logarítmicas multivariadas. Em seguida, propomos um modelo de regressão Birnbaum-Saunders generalizado multivariado. Métodos para estimação dos parâmetros do modelo, tal como o método de máxima verossimilhança baseado no algoritmo EM, foram desenvolvidos. Estudos de simulação de Monte Carlo foram realizados para avaliar o desempenho dos estimadores propostos. Segundo, realizamos uma análise de diagnóstico para modelos de regressão Birnbaum-Saunders generalizados multivariados. Consideramos a distância de Mahalanobis como medida de influência global de detecção de outliers multivariados utilizando-a para avaliar a adequacidade do modelo. Além disso, desenvolvemos medidas de diagnósticos baseadas em influência local sob alguns esquemas de perturbações. Implementamos a metodologia apresentada no software R, e ilustramos com dados reais multivariados de biomateriais. Terceiro, e finalmente, desenvolvemos uma metodologia robusta baseada em gráficos de controle de qualidade multivariados para a distribuição Birnbaum-Saunders generalizada usando a estatística de Hotelling. Baseado no método bootstrap paramétrico encontramos aproximações da distribuição desta estatística e obtivemos limites de controle para o gráfico proposto. Realizamos um estudo de simulação de Monte Carlo para avaliar a metodologia proposta indicando seu bom desempenho para fornecer alertas precoces de processos fora de controle. Uma ilustração com dados reais de qualidade do ar de Santiago-Chile é fornecida. Essa ilustração mostra que a metodologia proposta pode ser útil para alertar sobre episódios de poluição extrema do ar, evitando efeitos adversos na saúde humana.