  An important question in health services research is the estimation of the
proportion of medical expenditures that exceed a given threshold. Typically,
medical expenditures present highly skewed, heavy tailed distributions, for
which (a) simple variable transformations are insufficient to achieve a
tractable low-dimensional parametric form and (b) nonparametric methods are not
efficient in estimating exceedance probabilities for large thresholds.
Motivated by this context, in this paper we propose a general Bayesian approach
for the estimation of tail probabilities of heavy-tailed distributions, based
on a mixture of gamma distributions in which the mixing occurs over the shape
parameter. This family provides a flexible and novel approach for modeling
heavy-tailed distributions, it is computationally efficient, and it only
requires to specify a prior distribution for a single parameter. By carrying
out simulation studies, we compare our approach with commonly used methods,
such as the log-normal model and nonparametric alternatives. We found that the
mixture-gamma model significantly improves predictive performance in estimating
tail probabilities, compared to these alternatives. We also applied our method
to the Medical Current Beneficiary Survey (MCBS), for which we estimate the
probability of exceeding a given hospitalization cost for smoking attributable
diseases. We have implemented the method in the open source GSM package,
available from the Comprehensive R Archive Network.
