I am interested in the following practical question: Given two measures (say those of two parametric distributions), is there an algorithm for computing the Prokhorov metric between them?
The general definition of the Prokhorov metric is as follows. For two finite measures $\mu_1$, $\mu_2$ on a separable metric space $\left( X, d \right)$, that metric is defined as $ \rho \left( \mu_1, \mu_2 \right) = \inf \left\{ \varepsilon > 0 : \mu_1 \left( G \right) \leqslant \mu_2 \left( G^{\varepsilon} \right) + \varepsilon, \forall G \in \mathcal{B} \right\} $ where $\mathcal{B}$ is the Borel $\sigma$-algebra on $X$ and $G^{\varepsilon} = \left\{ x \in X : \inf_{y \in G} d \left( x, y \right) < \varepsilon \right\}$.
This metric is quite useful in the theory of weak convergence of probability measures on metric spaces (See Billingsley [Convergence of Probability Measures] or van der Vaart and Wellner [Weak Convergence and Empirical Processes]). The purpose of my question is that I am curious about whether a constructive algorithmic approach has been already studied. And if not, how could that be accomplished.