Transporting probability measures : some contributions to statistical inference

In statistics, one aims at understanding phenomena based on some data. Given the inherent variability of the observations at hand, the latter are often modelled as random variables whose study hinges on measure theory. Informally stated, a measure can be seen as a description of how probabilities are assigned to a collection of events. In such a framework, there exist a notion of transport of measures and a theory of optimal transport. The latter theory blossomed recently and its usefulness for statistical applications has been gradually acknowledged. This thesis is anchored in the domain of measure transportation and optimal transport, with a particular focus on statistical applications. In this context, we proposed and studied original procedures that address classical, although quite fundamental, issues of statistics. The thesis is mostly devoted to proposing a test for the two-sample problem, carrying out goodness-of-fit tests and measuring dependence between random vectors, while additional, corollary topics are presented as well. The first main problem aims at concluding that the multivariate datasets differ between two groups and was tackled via a particular transport of measure. The two subsequent core problems that we addressed relied on the optimal transport theory and particularly the so-called Wasserstein distance. Goodness-of fit tests are used to evaluate whether the data does not come from a given model, while the proposed dependence coefficients help understand how two multivariate quantities are linked. The concepts and tests developed in the thesis are thus quite general and aim at having wide applicability.