Calculation of the Prokhorov distance by optimal quantization and maximum flow

Calculating exact values of the Prokhorov metric for the set of probability distributions on a metric space is a challenging problem. In this paper probability distributions are approximated by finite-support distributions through optimal or quasi-optimal quantization, in such a way that exact calculation of the Prokhorov distance between a distribution and a quantizer can be performed. The exact value of the Prokhorov distance between two quantizers is obtained by solving an optimization problem through the Simplex method. This last value is used to approximate the Prokhorov distance between the two initial distributions, and the accuracy of the approximation is measured. We illustrate the method on various univariate and bivariate probability distributions. Approximation of bivariate standard normal distributions by quasi-optimal quantizers is also considered.