In statistical literature the Box-Cox transformations are used to make the data more suitable for statistical analysis. Considering the cases where the extreme data are strictly positive what is the effect of a Box-Cox power transformation on the data? We know from the literature that this transformation of the data can increase the rate of convergence of the tail of the distribution to the generalized extreme value distribution and as by product the bias of the estimation procedure is reduced. The reduction of bias of the Hill estimator has been extensively addressed in the literature of extreme value theory. Several techniques have been used to achieve such reduction of bias, either by removing the main component of the bias of the Hill estimator of the extreme value index (EVI) or by constructing new estimators based on generalized means or norms that generalize the Hill estimator. In this talk we are going to study the Box-Cox Hill estimator introduced in Teugles and Vanrolen, in 2004. We shall prove the consistency and asymptotic normality of the estimator and address the choice and estimation of the power and shift parameters of the Box-Cox transformation, not only for the EVI-estimation, but also for the estimation of other parameters of extreme events. The performance of the estimators under study will be illustrated for finite samples through small-scale Monte-Carlo simulation studies.

Joint work with M. Ivette Gomes