We are interested in reduced-bias versions of a simple generalisation of the classical Hill estimator of a positive extreme value index (EVI), the primary parameter of extreme events. The Hill estimator (Hill, 1975) can be regarded as the logarithm of the geometric mean of a set of statistics, dependent on a tuning parameter k related to the number of top order statistics involved in the estimation. Instead of such a geometric mean, we can more generally consider the mean of order p (MOP) of those statistics. The asymptotic behaviour of the class of MOP EVI-estimators, detailed in Brilhante et al. (2013), is reviewed, and associated reduced-bias MOP (RBMOP) and optimal RBMOP (ORBMOP) classes of EVI-estimators are suggested and studied both asymptotically and for finite samples. With PORT standing for peaks over random threshold, a terminology coined in Araújo Santos et al. (2006), we further advance with a PORT-RBMOP EVI-estimator, dependent on an extra tuning parameter q in [0,1). An adequate choice of the tuning parameters (k, q) is put forward, and an application to simulated and real data is performed.

References
[1] Araújo Santos, P., Fraga Alves, M.I. and Gomes, M.I. (2006). Peaks over random threshold methodology for tail index and high quantile estimation, Revstat 4:3, 227--247.
[2] Brilhante, M.F., Gomes, M.I. and Pestana, D. (2013). A simple generalization of the Hill estimator, Computational Statistics and Data Analysis 57:1, 518--535.
[3] Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution, Ann. Statist. 3, 1163--1174.