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 ^{− λ β} , 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 = λ β , 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 τ ₅ = λ β γ . 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.