Ordering claim size distributions and mixed Poisson probabilities

We investigate various orderings between continuous distributions for severities, having the same first n moments. Such situations occur for instance when severity distributions are fitted by the method of moments. l results are derived which establish an ordering between such distributions, and these results are applied to compare the Gamma, the Inverse Gaussian and the lognormal distributions with equal means and variances. Finally, we consider the situation where such continuous distributions are used as mixing distributions in mixed Poisson models for claim numbers, and show that the order properties of the mixing distributions are inherited by the corresponding mixed Poisson distributions.