PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 – 1999 ASYMPTOTIC DISTRIBUTION OF GUMBEL STATISTIC IN A SEMI-PARAMETRIC APPROACH (*) M.I. Fraga Alves Abstract: This note is an answer to some open problems connected with recent developments for appropriate methodologies for making inferences on the tail of a distri- bution function (d.f.). Namely, in Fraga Alves and Gomes (1996), the Gumbel statistic , based on the top part of a sample, is used in a semi-parametric approach, in order to fit an appropriate tail to the underlying model to a data set. The problem of statistical inference about extremal observations is handled there according to a test for choosing the most appropriate domain of attraction for the tail distribution, which gives prefer- ence to the Gumbel domain for the null hypothesis. The asymptotic behaviour of the referred statistic is derived therein under that null hypothesis and here we present similar extended results under the alternative conditions, i.e., for d.f. that belongs to the other Generalized Extreme Value domains, as an accomplishment to the promise made in last chapters of Fraga Alves and Gomes (1995; 1996). 1 – Introduction Suppose we are interested in making inferences about extremal values of some random variable, for which we have an available data set, in such a way that it is reasonable to identify it with X 1 ,X 2 , ..., X n , an independent, identically distributed (i.i.d.) sample from a d.f. F ( · ; λ, δ), where λ ∈ R and δ> 0 are eventually the location and the scale parameters, respectively. There has been several approaches to accomplish the main objective of in- ferring about very extremal values of the random quantity under research, from Received : November 11, 1997; Revised : May 22, 1998. AMS Subject Classification : 62E20, 62E25, 62G30, 26A12. Keywords : Extreme-value theory, order statistics, inference on the tail, regular variation, π-variation. (*) This research project was partially supported by MODEST - PRAXIS XXI and FEDER.