For estimating the distribution function F of a population, the empirical or sample distribution function Fn has been studied extensively. Qin and Lawless have proposed an alternative estimator &Fcirc;n for estimating F in the presence of auxiliary information under a semi-parametric model. They have also proved the point-wise asymptotic normality of &Fcirc;n. In this paper, we establish the weak convergence of &Fcirc;n to a Gaussian process and show that the asymptotic variance function of &Fcirc;n is uniformly smaller than that of Fn. As an application of &Fcirc;n, we propose to employ the mean &Xmacrcrc;n and variance Ŝ2n of &Fcirc;n to estimate the population mean and variance in the presence of auxiliary information. A simulation study is presented to assess the finite sample performance of the proposed estimators &Fcirc;n, &Xmacrcrc;, and Ŝ2n.
