Goodness--of--Fit Tests Based on the Min--Characteristic Function

We propose tests of fit for classes of distributions that include the Weibull, the Pareto and the Fr\'echet, distributions. The new tests employ the novel tool of the min--characteristic function and are based on an L2--type weighted distance between this function and its empirical counterpart applied on suitably standardized data. If data--standardization is performed using the MLE of the distributional parameters then the method reduces to testing for the standard member of the family, with parameter values known and set equal to one. We investigate asymptotic properties of the tests, while a Monte Carlo study is presented that includes the new procedure as well as competitors for the purpose of specification testing with three extreme value distributions. The new tests are also applied on a few real--data sets.