Nonparametric homogeneity tests

We test whether two independent samples of i.i.d. random variables X₁,...,X_n and Y₁,..., Y_m having common probability density f and, respectively, g are issued from the same population. The null hypothesis f = g is opposed to a large nonparametric class of smooth alternatives f and g. We consider several problems, according to the distance between the populations' densities: Point-wise, interval-wise, L₂ and L_∞ norms. We propose test procedures that attain parametric rates in some cases. In other problems, the procedures adapt automatically to the smoothnesses of the underlying densities. After a numerical study of these tests, we prove their theoretical properties in the classical minimax approach.