Moment diagnostics for NBUE/NWUE ageing of the conditional time to ruin in the classical Cramér–Lundberg risk model

We study the ageing behaviour of the conditional time to ruin in the classical Cramér–Lundberg risk model. The focus is not the ultimate ruin probability itself, but the distribution of the time at which ruin occurs conditional on ruin occurring. From the first two conditional moments we construct a reserve-dependent statistic Σ(u) which gives a rigorous exclusion test for the NBUE and NWUE ageing classes. For general light-tailed claim sizes satisfying a Cramér condition, defective-moment asymptotics imply Σ(u)→1/2 as the initial reserve u→∞, while explicit moment calculations give Σ(0)>1 under finite third moment and moderate loading. Thus a single exclusion conclusion cannot hold uniformly across all reserve levels. For exponential claim sizes, Σ(u) is obtained in closed form, is strictly decreasing, and yields an explicit reserve threshold at which the exclusion implication changes.