A new class of estimators of a 'scale' second order parameter

摘要

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.