A broad class of statistics including U-statistics and generalized quantiles (U- quantiles) can be viewed as solutions to an equation of the form. For these generalized M- estimators we develop weak and strong representations. As an application of the weak representation we show a central limit theorem for Mn. From the a.s. representation for mnthe asymptotic properties of sequential fixed width confidence intervals based on these estimators are derived. The results in this paper unify and extend earlier work of Geertsema (1970), Jurečková (1979), Boos (1980) and Choudhury and Serfling (1985).