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For a large class of heavy-tailed distribution functions F in the domain of attraction for maxima of an Extreme Value distribution with tail index γ > 0, the function A(t), controlling the speed of convergence of maximum values, linearly normalized, towards a non-degenerate limiting random variable, may be parameterized as A(t) = γ β t ρ , ρ < 0, β ∈ R, where β and ρ are second order parameters. The estimation of ρ, the "shape" second order parameter has been extensively addressed in the literature, but practically nothing has been done related to the estimation of the "scale" second order parameter β. In this paper, and motivated by the importance of a reliable β-estimation in recent reduced bias tail index estimators, we shall deal with such a topic. Under a semi-parametric framework, we introduce a class of β-estimators and study their consistency. We deal with the conditions enabling us to get the asymptotic normality of the members of this class, and we illustrate the behaviour of the estimators, through Monte Carlo simulation techniques.

Statistical theory of extremes, semi-parametric estimation, Primary 62G05, 62E25, 62E20, Secondary 62F35,

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation of the Weibull tail coefficient on the power mean-of-order- p . Consistency and asymptotic normality of the estimators under study are put forward. Their performance for finite samples is illustrated through a Monte Carlo simulation. It is always possible to find a negative value of p (contrarily to what happens with the mean-of-order- p estimator for the extreme value index), such that, for adequate values of the threshold, there is a reduction in both bias and root mean square error.