Spatial scan statistics based on empirical likelihood and robust fitting for generalized additive models for location, scale and shape

Abstract

This thesis presents two independent themes with different background. The first theme presents a new method for detecting spatial clusters, that is, a method for detecting regions with a high concentration of spatial phenomena, compared to a expected number, given a random distribution of events. The main contribution is to present a nonparametric method based on empirical likelihood functions, as an alternative to traditional methods of using clusters (scan). In this way, no distribution family is required for the variable of interest. To evaluate the method, simulation studies were carried out considering the zero-inflated poisson model, comparing the results with the scan method proposed by Kuldorff. The results show that the new method reduces the error probabilities of the type I for zero inflated, with low power for cluster with less than 8 locations. A study was carried out for Measles data in São Paulo, Brazil, which present a excess of zeros. Only the Kulldorff scanning method identified the existence of a cluster, located and centered on the capital São Paulo. However, if a cluster is identified by the Kulldorff method in the presence of inflated and when not confirmed by the non-parametric approach, it is recommended that interpretations be performed with caution due to a high probability type error associated with Kulldorff method when model is not well specified. The second theme aims to present two new approaches to robust estimation for generalized additive models of location, scale and shape - GAMLSS, which focus on contamination situations in the tails of distributions. The main motivation is the scarcity of robust methods for GAMLSS models. The thesis were subdivided into two topics. The first topic presents a proposal that seeks transformations in order to limit the influence function associated with the probability distribution of interest, modifying the logarithm structure of the likelihood function, using concepts of censorship. It also features: the robust GAMLSS method proposed by Rigby et al. (2019), considering the gamma distribution, presenting the bias corrections for the estimators; a modification of the method proposed by Rigby et al. (2019), considering the weight of observations in the estimation; and, finally, a large simulation study to evaluate the proposals, using the gamma distribution and contamination in the right tail of the distribution. The second topic 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. The simulation studies used the gamma and beta distributions, left and right tail contamination, and three distinct models: parametric models with and without 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 on acute viral infection of the respiratory system. excess of zeros. Only the Kulldorff scanning method identified the existence of a cluster, located and centered on the capital São Paulo. However, if a cluster is identified by the Kulldorff method in the presence of inflated and when not confirmed by the non-parametric approach, it is recommended that interpretations be performed with caution due to a high probability type error associated with Kulldorff method when model is not well specified.