Let x1:, x2, x:3, ... be a sequence of independent identically distributed random variables with unknown distribution function F. We want to estimate an estimable para-meter functionby the corresponding sequence unof U-statistics with loss function.For a given t > 0 there exists no fixed sample size guaranteeing that the risk Rn=E(Ln) is less than t. In analogy to CHOW ROBBINS (1965) asymptoti c consistency and asymptotic sample-size-efficiency are defined and an appropriate stopping time is given