1Universidad de Santiago de Chile, Facultad de Ciencia, Departamento de Matem\'atica y Ciencia de la Computaci\'on, Santiago, Chile. Assistant professor. Email: francisco.torres@usach.cl
2Universidad de Talca, Instituto de Matemática y F\'{\i}sica, Talca, Chile. Associate professor. Email: gicaza@utalca.cl
3Pontificia Universidad Cat\'olica de Chile, Facultad de Matem\'aticas, Departamento de Estad\'{\i}stica, Santiago, Chile. Professor. Email: reivalle@mat.puc.cl
This work considers distributions obtained as scale mixture of normal densities for correlated random variables, in the context of the Markov random field theory, which is applied in Bayesian spatial intrinsically autoregressive random effect models. Conditions are established in order to guarantee the posterior distribution existence when the random field is assumed as scale mixture of normal densities. Lung, trachea and bronchi cancer relative risks and childhood diabetes incidence in Chilean municipal districts are estimated to illustrate the proposed methods. Results are presented using appropriate thematic maps. Inference over unknown parameters is discussed and some extensions are proposed.
Key words: Disease mapping, Markov random field, Hierarchical model, Incidence rate, Relative risk.
Este trabajo aborda las distribuciones obtenidas como mezcla de escala de normales para variables aleatorias correlacionadas, en el contexto de la teor{\i}a de los campos markovianos, la cual es aplicada a modelos bayesianos espaciales con efectos aleatorios autoregresivos intr{\i}nsecos. Se establecen condiciones para garantizar la existencia de la distribuci\on a posteriori cuando se asume una distribuci\on mezcla de escala de normales para el campo markoviano propuesto. Para ilustrar los m\etodos propuestos, se estiman los riesgos relativos de c\ancer de tráquea, bronquios y pulm\on, y tasas de incidencia de diabetes tipo 1 en distritos municipales de Chile. Los resultados son presentados usando mapas tem\aticos apropiados. Se discute la inferencia sobre los par\ametros desconocidos y se proponen algunas extensiones.
Palabras clave: campo aleatorio markoviano, mapeo de enfermedades, modelo jerárquico, riesgo relativo, tasa de incidencia.
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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv35n2a01,
AUTHOR = {Torres-Avil\'es, Francisco J. and Icaza, Gloria and Arellano-Valle, Reinaldo B.},
TITLE = {{An Extension to the Scale Mixture of Normals for Bayesian Small-Area Estimation}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2012},
volume = {35},
number = {2},
pages = {185-204}
}