Consider two independent samples, one sample of size m from a distribution F and the other of size n from a weighted distribution G where   with w(.)≤0 and   Assume that there is a parameter θεR^d associated with F through   and consider the nonparametric estimators   of F and   of G on the basis of these two samples when θ is known and Φ is a real valued function and when θ is unknown and Φ is a rector valued function of dimension r<d. We show that   converge weakly to pinned Gaussian processes as m+n goes to +∞ and m/n converges to a constant and provide the expressions of the covariance functions. In the case where θ is unknown and Φ is a vector valued function of dimension r<d, we propose an approximate chi-square test for testing θ = θ₀ against all alternatives. This work is an extension of Vardi (1982a,b) and is closely connected to the work of Qin (1993) and Qin and Lawless (1995).