This thesis presents proposals for two independent themes and contributions to three different topics. The main ideas of each theme are presented in the next paragraphs.

The first topic accepted for publication presents a new method for detecting spatial clusters, that is, a method for detecting regions with a high concentration of spatial phenomena, compared with an expected number, given a random distribution of events. The main contribution of the proposal is to present a non-parametric method, based on empirical likelihood functions, as an alternative to traditional methods of cluster scan existing in the literature. Thus, no distribution family is required for the variable of interest. To evaluate the method, simulation studies were carried out considering the Poisson model inflated with zeros, comparing the results with the scan method proposed by Kuldorff. The results show that the new method reduces the type I error probabilities for zero-inflated observations, with low power for clusters with less than 8 locations. A measles case study in the region of São Paulo, Brazil was carried out. Observations have a high occurrence of zeros. Only the Kulldorff scan method identified the existence of a cluster, located and centered in the capital São Paulo. However, if a cluster is identified by the Kulldorff method in the presence of inflated observations and when not confirmed by the non-parametric approach, it is recommended that the interpretations be performed with caution due to the high probability of type I error associated with the Kulldorff method when the model does not is well specified.

 

The second theme aims to present topics with two new approaches to robust modeling for generalized additive models of location, scale and shape (GAMLSS). The main motivation is the scarcity of robust methods for GAMLSS models. Both proposals seek transformations in order to limit the influence function associated with the probability distribution of interest, and focus on situations of  contamination in the tails of the distributions. The first approach modifies the logarithm structure of the likelihood function, using concepts of censoring. Simulations studies were carried out to evaluate the methodology and applications are presented. The second approach in this theme is based on a simple adaptive truncation, where observations identified as possible outliers are verified and, if necessary, removed by truncation of the response variable distribution. In addition to proposing new robust modeling methods, they were compared with some methods already available in the literature. The simulation studies used the gamma and beta distributions, considering three distinct models: parametric models without and with covariates and non-parametric models. The results show that, compared to existing methods in the literature, the truncated adaptive method has a better performance with lower mean square error and lower variability in most simulated scenarios. The overall performances of the proposals are illustrated through three applications: brain image resonance data, using bivariate smoothing splines; extreme child poverty data; and data from severe acute respiratory syndrome - SRAG.