The contribution of the maximum to the sum of excesses for testing max-domains of attraction

We consider an i.i.d. sample, from an underlying distribution function with unknown shape, location and scale parameters, belonging to some max-domain of attraction. We study the performance of a test statistic which is merely a ratio between the maximum and the mean of the sample of the excesses above some random threshold. This scale/location invariant ratio turns out to be very useful in the construction of an asymptotically size αα test for the null hypothesis that the distribution comes from the Gumbel domain of attraction. The test is based on the knkn largest observations, where knkn is any intermediate sequence of positive integers. Both power of the test and type I error probability are studied for finite sample sizes by simulation.