Exponential functionals and means of neutral-to-the-right priors

The mean of a random distribution chosen from a neutral-to-the-right prior can be represented as the exponential functional of an increasing additive process.This fact is exploited in order to give sufficient conditions for the existence of the mean of a neutral-to-the-right prior and for the absolute continuity of its probability distribution. Moreover, expressions for its moments, of any order, are provided. For illustrative purposes we consider a generalisation of the neutral-to-the-right prior based on the gamma process and the beta-Stacy process. Finally, by resorting to the maximum entropy algorithm, we obtain an approximation to the probability density function of the mean of a neutral-to-the-right prior. The arguments are easily extended to examine means of posterior quantities. The numerical results obtained are compared to those yielded by the application of some well-established simulation algorithms.