Asymptotic distributions of linear and non linear combinations of extreme order statistics

Abstract

The limit distributions of linear and non-linear combinations of the kn = o(n) order statistics of i.i.d. random variables whose maximum belongs to the domain of attraction of the Gumbel law are obtained. Our results may be applied in actuarial studies, estimation of scale-location parameters, estimation of squared deviation in tail of a distribution, robustness theory and detection of the outliers in statistical data. It is also closely related to the moment estimator of Dekkers-Einmahl-de Hann (1989) for the index of an extreme distribution. This study completes that of Necir (1990, 1991a, 1991b, 2000a, 2000b).