Effective Reparameterized Importance Sampling for Spatial Generalized Linear Mixed Models with Parametric Links
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Spatial generalized linear mixed models (SGLMMs) have been popular for analyzing non-Gaussian spatial data observed in a continuous region. These models assume a prescribed link function that relates the underlying spatial random field with the mean response. On the other hand, there are circumstances, such as when the data contain outlying observations, where the use of a prescribed link function can result in a poor fit which can be improved by the use of a parametric link function. In this paper we present different sensible choices of parametric link functions which possess certain desirable properties. It is important to estimate the parameters of the link function, rather than assume a known value. To that end, we present a generalized importance sampling (GIS) estimator based on multiple Markov chains for an empirical Bayes analysis of SGLMMs. It turns out that the GIS estimator, although more efficient than simple importance sampling, can be highly variable when it is used to estimate the parameters of certain link functions. We propose two modified GIS estimators based on suitable reparameterizations (transformations) of the Monte Carlo samples. These transformations are also used to eliminate the well-known separability problem of Geyer's (1994) reverse logistic regression estimator. We also provide a new method based on Laplace approximation for choosing the multiple importance densities (or skeleton points) in the GIS estimator. Finally, we discuss a methodology for selecting models with appropriate link function family, which extends to choosing a spatial correlation function as well. The proposed estimators and methodology are illustrated using both simulation and real data examples.
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