  Insurance data can be asymmetric with heavy tails, causing inadequate
adjustments of the usually applied models. To deal with this issue,
hierarchical models for collective risk with heavy-tails of the claims
distributions that take also into account overdispersion of the number of
claims are proposed. In particular, the distribution of the logarithm of the
aggregate value of claims is assumed to follow a Student-t distribution.
Additionally, to incorporate possible overdispersion, the number of claims is
modeled as having a negative binomial distribution. Bayesian decision theory is
invoked to calculate the fair premium based on the modified absolute deviation
utility. An application to a health insurance dataset is presented together
with some diagnostic measures to identify excess variability. The variability
measures are analyzed using the marginal posterior predictive distribution of
the premiums according to some competitive models. Finally, a simulation study
is carried out to assess the predictive capability of the model and the
adequacy of the Bayesian estimation procedure.
  Keywords: Continuous ranked probability score (CRPS); decision theory;
insurance data; marginal posterior predictive; tail value at risk; value at
risk.
