  Case-deleted analysis is a popular method for evaluating the influence of a
subset of cases on inference. The use of Monte Carlo estimation strategies in
complicated Bayesian settings leads naturally to the use of importance sampling
techniques to assess the divergence between full-data and case-deleted
posteriors and to provide estimates under the case-deleted posteriors. However,
the dependability of the importance sampling estimators depends critically on
the variability of the case-deleted weights. We provide theoretical results
concerning the assessment of the dependability of case-deleted importance
sampling estimators in several Bayesian models. In particular, these results
allow us to establish whether or not the estimators satisfy a central limit
theorem. Because the conditions we derive are of a simple analytical nature,
the assessment of the dependability of the estimators can be verified routinely
before estimation is performed. We illustrate the use of the results in several
examples.
