Abstract
Hydrological and cryospheric risk assessment depends on reliable inference from scarce but high-impact observations, yet classical tail-index estimators can be sensitive to threshold choice, location shifts, contamination, and very small effective sample sizes. This paper develops and evaluates a family of median-normalized tail-index estimators for heavy-tailed distributions in the Fréchet domain of attraction. Threshold exceedances are normalized by their sample median and transformed using either log r or log (1 + r), after which the tail index is recovered through calibrated inverse maps. The construction is affine invariant by design. We establish the probability limit of the median exceedance, prove global invertibility of the two calibration maps, and obtain consistency under regular variation. For independent exact generalized-Pareto (GPD) exceedances, we establish a joint central limit theorem for LMC1 and LMC2 and transfer this limit exactly to upper-order statistics under an exact Pareto model. For general non-exact regularly varying tails, the corresponding second-order joint limit requires additional bias and excess-process conditions and is not claimed here.
The empirical application uses annual glacier mass-balance observations from Echaurren Norte and ten additional Chilean glacier-loss series included in the comparative analysis. Because the real-data tail index is unknown, the application is explicitly exploratory and focuses on threshold sensitivity, estimator admissibility, and agreement among classical and median-normalized estimators rather than claims of optimality. For Echaurren Norte, the glacier-loss component provides the most informative tail sample and shows marked dependence of estimates on the number of upper-order observations. Across the additional glaciers, simultaneous admissibility of the two median-normalized estimators occurs only in sample-specific windows, with no universal value of k. These findings support a cautious workflow in which admissibility, threshold sensitivity, and conventional extreme-value diagnostics are assessed jointly. The paper concludes with priorities for simulation-based validation, second-order theory, and hydrological interpretation.