Detecting finiteness in the right endpoint of light-tailed distributions

In Extreme Value statistics we often encounter testing procedures for assessing the presence of the Gumbel domain, attached to the simple null hypothesis of shape parameter γ=0γ=0, thus praising the selection of extreme domains of attraction. However, the problem of assessing for light-tailed distributions with finite or infinite right endpoint is seldom referred. The latter is an impending problem of practical importance, particularly at the enrollment of subsequent estimation of extremal features such as small exceedance probabilities. In this paper, we present two test statistics whose asymptotic behavior, albeit under some restrictive yet reasonable conditions, enables us to distinguish light-tailed distribution functions with finite right endpoint from those with infinite endpoint lying in the Gumbel domain. An illustrative example is provided via application to significant wave height data recorded at Figueira da Foz, Portugal, from 1958 until 2001.