El propósito de estos seminarios es conocer los proyectos de investigación en los que han participado los y las estudiantes del programa de Doctorado en Estadística en la modalidad de ponencia. Se extiende la invitación a participar a toda la comunidad UC.
The choice of a prior distribution is a key aspect of the Bayesian method. However, in many cases, such as the family of power links, this is not trivial. In this article, we introduce a penalized complexity prior (PC prior) of the skewness parameter for this family, which is useful for dealing with imbalanced data. We derive a general expression for this density and show its usefulness for some particular cases such as the power logit and the power probit links. A simulation study and a real data application are used to assess the efficiency of the introduced densities in comparison with the Gaussian and uniform priors. Results show improvement in point and credible interval estimation for the considered models when using the PC prior in comparison to other well-known standard priors.
Scale-free networks play a fundamental role in the study of complex networks and various applied fields due to their ability to model a wide range of real-world systems. A key characteristic of these networks is their degree distribution, which often follows a power-law distribution, where the probability mass function is proportional to $x^{-\alpha}$, with $\alpha$ typically ranging between $2 < \alpha < 3$. In this talk, we introduce Bayesian inference methods to obtain more accurate estimates than those obtained using traditional methods, which often yield biased estimates, and precise credible intervals. Through a simulation study, we demonstrate that our approach provides nearly unbiased estimates for the scaling parameter, enhancing the reliability of inferences. We also evaluate new goodness-of-fit tests to improve the effectiveness of the Kolmogorov-Smirnov test, commonly used for this purpose. Our findings show that the Watson test offers superior power while maintaining a controlled type I error rate, enabling us to better determine whether data adheres to a power-law distribution. Finally, we propose a piecewise extension of this model to provide greater flexibility, evaluating the estimation and its goodness-of-fit features as well. In the complex networks field, this extension allows us to model the full degree distribution, instead of just focusing on the tail, as is commonly done. We demonstrate the utility of these novel methods through applications to two real-world datasets, showcasing their practical relevance and potential to advance the analysis of power-law behavior.
In the search for multivariate distributions that provide greater flexibility in modeling data characterized by high levels of skewness, kurtosis, and the presence of outliers, new families of multivariate distributions have emerged, among which multivariate normal mixture distributions stand out. In this context, we introduce a multivariate normal mixture distribution based on the Birnbaum-Saunders distribution and examine some of its key properties. To estimate the parameters of this normal scale mixture distribution, we propose a maximum likelihood approach implemented via the EM algorithm. To support inferential analyses, we derive the Fisher information matrix. Additionally, we formulate a linear hypothesis on the parameter vector of interest and evaluate it using the likelihood ratio, Wald, score, and gradient statistics. Finally, we illustrate the application of the proposed methodology to real datasets, complementing the analysis with a simulation study to assess its performance.