A random locational M-estimation problem based on the L2-Wasserstein distance

The fair placement of a facility often depends on other existing players and on an optimal assignment map of clients to these facilities. This problem arises in various contexts in decision mathematics such as, for instance, location theory in operational research and the interdisciplinary area of regional science. The random nature of client sites is implemented in our location-design model by relying on the Monge-Kantorovich mass transference problem. The criterion function to minimize is the Wasserstein distance between the unknown source mass distribution of clients and the partially known target mass distribution of facilities. A class of strongly consistent estimators of the optimal location and the best capacity constraint for the new facility is proposed. These estimators are shown to achieve the promising root-n speed of convergence. Explicit characterizations of both population and empirical optimal solutions are provided. There is in particular an interesting connection with quantile theory when the capacity constraint is prescribed, leading to the asymptotic normality of the optimal location estimator. Some simulation evidence is presented to evaluate the finite sample performance of the proposed estimators, where encouraging results are obtained. The ideas are also illustrated via efficient positioning of new monitoring stations in order to reinforce border controls between neighboring countries.