A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.
{Summary: We first consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parametrised in a tuning or control parameter $\alpha$. Such a control parameter enables us to have access for any available sample, to an estimator of the tail index $\gamma$ with a null dominant component of asymptotic bias and consequently with a reasonably flat mean squared error pattern, as a function of $k$, the number of top-order statistics considered.\par Such a control parameter depends on a second-order parameter $\rho$, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro-US\$ exchange rates.}