Consistency of objective Bayes factors for nonnested linear models and increasing model dimension

Casella et al. [2, (2009)] proved that, under very general conditions, for normal linear models the Bayes factor for a wide class of prior distributions, including the intrinsic priors, is consistent when the number of parameters does not grow with the sample size n. The special attention paid to the intrinsic priors is due to the fact that they are nonsubjective priors, and thus accessible priors for complex models. The case where the number of parameters of nested models grows as O(nα) for α ≤ 1 was considered in Moreno et al. [13, (2010)], in which it was proved that the Bayes factor for intrinsic priors is consistent for the case where both models are of order O(nα) for α < 1, and for α = 1 is consistent except for a small set of alternative models. The small set of models for which consistency does not hold was characterized in terms of a pseudo-distance between models. The goal of the present article is to extend the above results to the case where the linear models are nonnested. As the comparison of nonnested models calls for a method of encompassing, for proving consistency we use encompassing from below in this paper. Consistencia de factores de Bayes objetivos para modelos lineales anidados cuando la dimensión de los modelos crece Resumen. En Casella et al. [2, (2009)] se demostró que, bajo condiciones muy generales, el factor de Bayes para modelos lineales normales y para una amplia clase de distribuciones a priori, que incluı́a a las a priori intrı́nsecas, es consistente cuando el número de parámetros no crece cuando lo hace el tamaño muestral n. Se prestó especial atención a las distribuciones a priori intrı́nsecas debido a que son distribuciones a priori no subjetivas y, por consiguiente, se pueden aplicar a modelos complejos. El caso en que el número de parámetros de los modelos anidados crece del orden de O(nα) para α ≤ 1 se ha considerado en Moreno et al. [13, (2010)], en el que se demuestra que el factor de Bayes para distribuciones intrı́nsecas es consistente para el caso en que ambos modelos son de orden O(nα) para α < 1 y, para el caso α = 1, también es consistente excepto para un conjunto pequeño de modelos alternativos. Este conjunto, para el cual la consistencia no se da, se caracterizó en términos de una pseudo distancia entre modelos. Submitted by David Rı́os Received: December 28, 2009. Accepted: February 3, 2010