A general estimator for the right endpoint with an application to supercentenarian women’s records
We tackle the problem of right endpoint estimation in extremes using a recent semi-parametric estimator envisioned for distributions in the Gumbel domain, one of the only possible three max-domains of attraction induced by the extreme value theorem. The Gumbel domain plays a central role in extreme value statistics as it includes distributions ranging from moderately heavy-tailed with infinite endpoint, to light-tailed distributions underlying extreme magnitudes which are bounded from above, i.e. with finite right endpoint. The Weibull max-domain of attraction is the remainder possibility for distributions with an upper bound. The scope of the proposed estimator is then lengthened to the Weibull domain, thus enabling a general estimator for the finite right endpoint. The main advantage of the general endpoint estimator is that it does not require external estimation of the (supposedly non-positive) extreme value index. We also show that it is strongly consistent but not always asymptotically normal since this endpoint estimator can exhibit different rates of convergence across the same domain of attraction. The finite sample performance is evaluated by means of Monte Carlo simulations. These convey that the general endpoint estimator works remarkably well in case the true extreme value index is greater than -1/2, which encapsulates the most common cases in practical applications. Finally, an illustration is provided through a brief analysis of supercentenarian women data, aiming at providing sensible bounds for the upper limit of the human life span (the ultimate human longevity).
Keywords and phrases: Extreme value theory, Regular variation, Semi-parametric estimation, Tail estimation, Human lifespan.