Tables are given for the asymptotic distribution function of the extremal quotient, which is valid for a wide class of initial distributions. Selected values of the shape parameter $\lambda$ are considered. This parameter depends upon the initial distribution, and the size of the sample from which the quotient is drawn. This distribution function is not too different from that of a logarithmic normal distribution. The large sample approach of the distribution function to the logistic one is very slow. A Monte Carlo study shows an unexpectedly good fit to the theory.
E. J. Gumbel. James Pickands III. "Probability Tables for the Extremal Quotient." Ann. Math. Statist. 38 (5) 1541 - 1551, October, 1967. https://doi.org/10.1214/aoms/1177698708