Simulated annealing for the bounds of Kendall's τ and Spearman's ρ

It has long been known that, for many joint distributions, Kendall's tau and Spearman's rho have different values, as they measure different aspects of the dependence structure. Although the classical inequalities between Kendall's tau and Spearman's rho for pairs of random variables are given, the joint distributions which can attain the bounds between Kendall's tau and Spearman's rho are difficult to find. We use the simulated annealing method to find the bounds for rho in terms of tau and its corresponding joint distribution which can attain those bounds. Furthermore, using this same method, we find the improved bounds between tau and rho, which is different from that given by Durbin and Stuart.