  In general insurance, claims are often lower-truncated and right-censored
because insurance contracts may involve deductibles and maximal covers. Most
classical statistical models are not (directly) suited to model lower-truncated
and right-censored claims. A surprisingly flexible family of distributions that
can cope with lower-truncated and right-censored claims is the class of MBBEFD
distributions that originally has been introduced by Bernegger (1997) for
reinsurance pricing, but which has not gained much attention outside the
reinsurance literature. Interestingly, in general insurance, we mainly rely on
unimodal skewed densities, whereas the reinsurance literature typically
proposes monotonically decreasing densities within the MBBEFD class. We show
that this class contains both types of densities, and we extend it to a bigger
family of distribution functions suitable for modeling lower-truncated and
right-censored claims. In addition, we discuss how changes in the deductible or
the maximal cover affect the chosen distributions.
