On approximating distributions by approximating their De Pril transforms

Dhaene & De Pril (1994) gave some general bounds for errors caused by applying an approximation to the De Pril transform of a probability distribution. In this paper we further analyse such bounds. We prove results for several simple situations and indicate how these results can be combined for more complex situations. As an illustration we apply the results to deduce approximations to the convolution of a finite number of compound distributions whose counting distributions belong to a class consisting of the binomial, Poisson, and negative binomial distributions, and we give error bounds for these approximations.