A Log-Exponential Pareto Distribution for Heavy-Tailed Insurance Data

Pareto-type models remain the workhorse of heavy-tailed loss modelling, but the classical Pareto distribution fixes a single tail decay rate that is often too restrictive for real claims data. We construct a new three-parameter extension, the N3PP distribution, by passing the Pareto survival function through a log-exponential transform, and derive its density, hazard, quantile function, moment-existence condition, and Shannon entropy in closed form. A tail index result shows that the upper tail decays as x−λβ, so the new shape parameter β rescales the Pareto tail rather than merely relocating it, and we show that this same tail behaviour governs a set of reinsurance-relevant quantities—the limited expected value, the mean excess function, and the integrated tail (equilibrium) distribution that links directly to classical ruin theory. A simulation study shows that λ and β are individually hard to pin down from finite samples, essentially because the Fisher information matrix is close to singular, but that their product τ=λβ, which is what actually governs the tail, is recovered with reasonable accuracy once the sample size is moderate. Adding a Marshall–Olkin frailty parameter γ and resilience parameter δ gives a five-parameter extension, N5PP, whose tail index is τ5=λβγ. We fit N5PP to the Swedish motorcycle insurance and Bragg claim datasets, compare it against five Pareto-type competitors, and compute value at risk and conditional tail expectation for each.