Nonparametric estimation in selection biased models in the presence of estimating equations

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 G(y) = 1/w integral(-infinity)(y) w(u)dF(u) with w(.) greater than or equal to 0 and 0 < w = integral(-infinity)(infinity) w(u)dF(u) < infinity. Assume that there is a parameter theta is an element of R-d associated with F through E(F)psi(x, theta) = 0 and consider the nonparametric estimators (F) over cap of F and (G) over cap of G on the basis of these two samples when theta is known and psi is a real valued function and when theta is unknown and psi is a vector valued function of dimension r > d. We show that root n+m((F) over cap - F) and root n+m((G) over cap - G) converge weakly to pinned Gaussian processes as m + n goes to + infinity and m/n converges to a constant and provide the expressions of the covariance functions. In the case where theta is unknown and psi is a vector valued function of dimension r > d, we propose an approximate chi-square test for testing theta = theta(0) 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).