The data-driven extreme value distribution: non-parametric tail estimation with a derived stability criterion

Abstract

Quantifying the likelihood of extreme events underpins risk assessment, yet classical Extreme Value Theory relies on asymptotic assumptions that fail in the data-sparse, non-stationary regimes practitioners increasingly face. We introduce the Data-Driven Extreme Value Distribution (DDEVD), a non-parametric estimator that aggregates all observations metastatistically and reconstructs the base distribution with a kernel, removing parametric tail assumptions. We derive its optimal bandwidth and prove a stability law $m<C{n}^{1+{\hat{I}}^3/2}$ relating reliable extrapolation to the extreme value index ${\hat{I}}^3$. In sub-hourly Alpine precipitation, DDEVD recovers stable 100-year return levels from single decades (calibration ratio $0.96$), departing from the full-record reference by over $50\%$ in fewer than one window in fifty -- versus one in five for a GEV fit. In metallurgical micrographs, it matches a generalised extreme-value fit on the safety-relevant grain-size tail, where the standard log-normal over-predicts by $58\%$ at $1{\mathrm{cm}}^2$.

28 pages, 6 figures