  A sequence of accompanying laws is suggested in the limit theorem of B. V.
Gnedenko for maximums of independent random variables belonging to maximum
domain of attraction of the Gumbel distribution. It is shown that this sequence
gives an exponential power rate of convergence whereas the Gumbel distribution
gives only a logarithmic rate. As examples, classes of Weibull and log-Weibull
type distributions are considered in details. A scale for the Gumbel maximum
domain of attraction is suggested as a continuation of the considered two
classes.
