Bayesian robust models with applicatións to small area estimatión.

Abstract

This work takes up methods for bayesian inference in generalized linear mixed models with applications to small-area estimation. A previous work (Datta and Lahiri, 1995) focused on Bayesian estimation with a prior scale mixture distribution for the error component in a normal linear model, to smooth small area means when one or more outliers are present in the data. Following this idea, an appropriate scale mixture of normals (Andrews and Mallows, 1974, Fernández and Steel, 2000) for the spatial random effects distribution is proposed in the context of the Markov random field theory, which is applied to the usual spatial intrinsically autoregressive random effect. Conditions are stablished in order to guarantee the posterior distribution existence when the random fiel d is observed directly. Given a joint observed random field, a simulation study is performed to illustrate the use of hierarchical algorithms. Inference over the variability parameter is obtained, showing that the best estimators are related to a particular scale mixture of normal random field. Based on the work of Ghosh et al. (1998), theoretical conditions are presented to guarantee the posterior distribution propriety, when a generalized linear mixed model with a spatial component is assumed. Due to the equivalence between the normal and the scale mixture of normal models, specifically with Student-t and Slash distributions, it is possible to obtain hierarchical representations, therefore, Markov Chain Monte Carlo sampler methods are used to perform the computations. Lung, trachea and bronchi cancer relative risk and childhood diabetes incidence in Chilean communes are estimated to illustrate the proposed methods. Interference over un-known parameters are discussed. Results are prsented using appropriate thematic maps. As part of the work in progress, theoretical aspects to measure Bayesian learning are explored, taking into account that in the spatial hierarchical model considered in this work, only the sum of two sets of random effects are identified by the data. Specific expressions for the L1 distance were obtained. Other considerations are discussed as part of the future work, considering extensions to be developed in different directions.