Asymptotic Distribution of Gumbel Statistic in a Semi-Parametric Approach

Maria Isabel 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 distribution 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 preference 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).

Keywords: Extreme-value theory; order statistics; inference on the tail; regular variation; $\pi$-variation.

Classification (MSC2000): 62E20, 62E25, 62G30, 26A12.

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