Nonparametric homogeneity tests

We test whether two independent samples of i.i.d. random variables X1,…,XnX1,…,Xn and Y1,…,YmY1,…,Ym having common probability density f and, respectively, g   are issued from the same population. The null hypothesis f=gf=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, L2L2 and L∞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.