Distribution of big claims in a Lévy insurance risk process: Analytics of a new non-parametric estimator

In this study, we model aggregate claims using a subordinator, specifically a non-decreasing Lévy process. Large positive jumps, exceeding a predetermined threshold, represent significant claims, while frequent but smaller fluctuations capture other sources of non-insurance uncertainty, such as miscellaneous expenses. The primary challenge lies in extracting the necessary mathematical insights to estimate the jump measure from a sample path of truncated aggregate claims. Through a discrete time-point sampling scheme, we conduct an initial comparison between conventional parametric estimators of the Lévy measure associated with the subordinator, based on simulated significant claims, and our proposed non-parametric estimator, derived by adapting classical differential processes originally introduced by Rubin and Tucker. The results of this comparison suggest the potential utility of our estimator in the context of real data from the insurance sector. While the primary focus of this work is to uncover the mathematical foundations, a preliminary simulation study, although lacking rigorous numerical analysis, hints at the favorable estimation of the Poisson rate for the number of jumps exceeding the threshold, achieved using our proposed non-parametric estimator of the Lévy measure.