BayesianSAE using hierarchical bayes.pdf

Inclusion of spatial random effects with a car

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Inclusion of spatial random effects with a CAR structure ( v i ), in combina- tion with unstructured random effects, tends to reduce bias and mean square error of the small area estimates compared to models with only unstructured or only spatial random effects. Models including spatial random effects based on the distance between areas ( w i , equation 10) do not perform well compared to the models with spatial random effects based on the CAR specification (re- sults not shown). In practice, the performance is very similar to the model with unstructured random effects u i . We believe that this happens because the stationarity assumption underlying the former specification does not hold. Another drawback of using this model is that fitting takes much longer than with other spatial models. Table 1 also shows that, broadly speaking, the ranking of models by DIC is similar to that based on MARB or MRRMSE, indicating that DIC can be useful for model selection in a real application setting (although note that the latter cannot be used to compare models based on different data, i.e. area versus unit level models). The right half of Table 1 summarises the bias and mean square error of the variance estimates from the different models. For all models, the variance estimates have much higher relative bias and mean square error than do the small area estimates themselves. Nevertheless, most of the Bayesian variance estimates (except unit level model 1 and the unit level model 2 with only spa- tial random effects) are substantially better than the corresponding synthetic variance estimates, and comparable or better than the variance of the direct estimator. In contrast to the findings for the small area point estimates, how- ever, the Bayesian models with spatial random effects only tend to perform poorly relative to models including unstructured random effects (either alone or in combination with spatial effects). The variance estimates from the former models tend to be systematically smaller than those from the latter (data not shown), suggesting that while borrowing of information from neighbouring areas can improve the small area estimates, it tends to over-estimate their precision. Thus models with both unstructured and spatial random effects may represent the best compromise in terms of producing accurate small area estimates which
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20 V. G´omez-Rubio et al. also have reasonable variance estimates. As an alternative way of assessing the variability of the small area estimates we have also computed frequentist coverage rates for all these models. Rao (2003, Chapter 10) discusses these issues and points out that posterior vari- ances tend to underestimate the frequentist MSE when the number of areas and/or the between-area variance is small, which may lead to too narrow cred- ible intervals. However, our results in Table 1 show that the coverage rates for the various Bayesian models considered here are only slightly lower than the nominal 95% in most cases. The exceptions are the Bayesian models with only
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  • Spring '16
  • Yessi
  • Regression Analysis, Mean squared error, Bayesian inference, Bayesian statistics

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