On some compound distributions with Borel summands

TL;DR

This paper introduces new compound distributions with Borel summands, including formulas and recursive structures, useful for modeling claim sizes in actuarial science, derived through probabilistic and combinatorial methods.

Contribution

It provides closed-form formulas and recursive structures for generalized compound distributions with Borel summands, extending existing models in actuarial applications.

Findings

Derived formulas for compound Bartlett and Delaporte distributions with Borel summands

Presented recursive structures for certain compound shifted Delaporte mixtures

Developed simple Panjer-type recursion formulas for claim size modeling

Abstract

The generalized Poisson distribution is well known to be a compound Poisson distribution with Borel summands. As a generalization we present closed formulas for compound Bartlett and Delaporte distributions with Borel summands and a recursive structure for certain compound shifted Delaporte mixtures with Borel summands. Our models are introduced in an actuarial context as claim number distributions and are derived only with probabilistic arguments and elementary combinatorial identities. In the actuarial context related compound distributions are of importance as models for the total size of insurance claims for which we present simple recursion formulas of Panjer type.