A projection type distribution function and quantile estimates in the presence of auxiliary information

The strong consistency, asymptotic normality and the law of the iterated logarithm of a projection type distribution function and quantile estimates in the presence of the auxiliary information Eg(X)=0 are obtained by using the empirical likelihood method. The Bahadur representation of a projection type quantile estimate is also given. Moreover, their asymptotic variances are smaller than that of classical distribution and quantile estimates, respectively.