On mean scaled insurance risk models

Abstract

A characterization problem for scale compound parametric families of distributions with the mean as scale parameter is analyzed. If the counting distribution of the compound family has the sample mean as maximum likelihood estimator of the counting mean, and if the maximum likelihood estimator of the mean scale parameter is the sample mean, then the compound family has necessarily a gamma secondary distribution. Necessary and sufficient conditions under which such a characterization holds are derived. Among the rich class of counting statistical models, which lead to such a characterization, one finds the Poisson, binomial, negative binomial, Hermite, Delaporte, extended Poisson-Pascal, mixed Poisson inverse Gaussian, Sichel, generalized Euler, and many others. The mixed Poisson lognormal is a counterexample for which the present characterization fails.