On the relation between the order of claims arrival and the ruin probability in a nonhomogeneous risk process

Recently, Raducan et al. (2014a) obtained recursive formulas for the ruin probabilityof a risk process at or before claim instants under the assumptions that the claim sizesare independent, nonhomogeneous Erlang distributed, and independent of the inter-claimtimes (i.e., the times between two successive claims), which were assumed to be inde-pendent, identically distributed (i.i.d.), following an Erlang or a mixture of exponentialsdistribution. Further on, Raducan et al. (2014b) extended these formulas to the moregeneral case when the inter-claim times are i.i.d. nonnegative random variables followingan arbitrary distribution. In this work, we present several numerical examples in whichwe vary the distribution of the inter-claim times, which, apart the two particular dis-tributions from Raducan et al. (2014a), is chosen of discrete and of uniform type. Acomparison with similar existing formulas in the literature for the homogeneous case isdiscussed. Moreover, all the examples for the nonhomogeneous case support a conjecturethat relates the order of the claims arrival with the magnitude of the corresponding ruinprobabilities. We also present the proof of a particular case of this conjecture, when theclaim sizes are nonhomogeneous exponentially distributed.