A hybrid-Hill estimator enabled by heavy-tailed block maxima

When analysing extreme values, two alternative statistical approaches have historically been held in contention: the seminal block maxima method (or annual maxima method, spurred by hydrological applications) and the peaks-over-threshold. Clamoured amongst statisticians as wasteful of potentially informative data, the block maxima method gradually fell into disfavour whilst peaks-over-threshold-based methodologies were ushered to the centre stage of extreme value statistics. This paper proposes a hybrid method which reconciles these two hitherto disconnected approaches. Appealing in its simplicity, our main result introduces a new universal limiting characterisation of extremes that eschews the customary requirement of a sufficiently large block size for the plausible block maxima-fit to an extreme value distribution. We advocate that inference should be drawn solely on larger block maxima, from which practice the mainstream peaks-over-threshold methodology coalesces. The asymptotic properties of the promised hybrid-Hill estimator herald more than its efficiency, but rather that a fully-fledged unified semi-parametric stream of statistics for extreme values is viable. A finite sample simulation study demonstrates that a reduced-bias off-shoot of the hybrid-Hill estimator fares exceptionally well against the incumbent maximum likelihood estimation that relies on a numerical fit to the entire sample of block maxima.