Multivariate regression models with measurement errors under generalized hyperbolic distributions
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2025
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Abstract
Measurement error models (MEMs), also known as errors-in-variables models, are useful tools used to describe various phenomena across multiple disciplines. MEMs establish functional relationships among variables that are observed with random measurement errors. This thesis focuses on studying the multivariate linear regression model when covariates are measured with error (MRMMEs),extending the classical multivariate linear regression framework. It is important to note that calibration models (CCMs) are a sub-class of MRMMEs. Then, we also study an extension of CCMs.Although the normality assumption is useful in some cases, the multivariate normal distribution is not appropriate when the data comes from asymmetric or heavy-tailed distributions. In practice, multivariate data are often complex in the sense that they present high levels of asymmetry, kurtosis, and outliers. For these reasons, the study of probabilistic models and inferential aspects for continuous multivariate observations is of interest. Then, this thesis aims to carry out a statistical analysis of the multivariate regression model with measurement error by considering the multivariate normal and distributions belonging to the multivariate generalized hyperbolic (GH) family.One of the significant challenges we face was dealing with identifiability issues, particulary within the multivariate GH family of distributions. To address this, we focused on the symmetric case, which includes the symmetric normal inverse Gaussian (sym NIG) distribution and the symmetric hyperbolic (sym HYP) distribution. We used the Maximum Likelihood (ML) framework for estimation through the Expectation Maximization algorithm, hypothesis testing, model assessment, and influence diagnostics in both the multivariate regression model with measurement error under the normality assumption (N-MRMME) and the multivariate regression model with measurement error under multivariate symmetric generalized hyperbolic distributions (sym GH-MRMME). In both cases, we faced the challenge of complex expressions. However the suitable properties of the multivariate normal distribution and the GH distributions simplified the process.All proposed methodologies were illustrated with real data applications. Additionally, simulation studies based on the applications were conducted to assess the behavior of the estimation procedures, hypothesis tests, and influence diagnostics, providing support for its validity. The results showed accurate estimations that improved with larger sample sizes, as expected. Even in the presence of high levels of kurtosis, the symmetric GH models presented a better fit than the N-MRMME.Ultimately, this work contributes to the existing literature on measurement error models and generalized hyperbolic distributions.
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Tesis (Doctor en Estadísticas)--Pontificia Universidad Católica de Chile, 2025