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.
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