An asymmetric test on the type of the distribution of extremes

The problem may be introduced by looking at fig. 1, in which (some points of) the empirical distribution of a sample of size 47 has been plotted on Gumbel probability paper. The data are annual maxima of discharges of the North Saskatchewan River at Edmonton, Canada, see table 3. In the same figure the maximum likelihood fitted distribution functions have been drawn assuming that the variate itself (I) or its logarithm (II) has a double exponential distribution. Except in the right tail the fitted distribution functions are nearly the same, but for extrapolation purposes in the right tail (a favourite activity in e.g. operational hydrology) there is a big difference. In this case the .99-points of the fitted distribution functions differ a factor 2.