Exponential and scale mixtures and equilibrium distributions

Abstract

In this article we discuss mixed exponential distributions and, more generally, scale mixtures with specific consideration the purpose of insurance modeling. Results are derived for equilibrium distributions (defined via stop-loss transforms) of mixed distributions. Some recursive relations are identified for the stop-loss transforms and moments of mixed exponential distributions. Explicit expressions are obtained for equilibrium gamma distributions with arbitrary shape parameter.

Key words:

• Mixed exponential
• equilibrium distributions
• excess-of-loss
• recursions