Deficiencies of u-statistics of degree 2 under symmetric distributions

For a continuous distribution with a certain symmetry, several U-statistics of degree 2 are asymptotically as efficient as the invariant U-statistics which are UMVU estimators of estimable parameters. To see the difference between two the statistics, we evaluate the limiting risk deficiency of the U-statistic with respect to the invariant U-statistic, which is also equal to the coefficient of the reciprocal of the sample size in the ratio of their variances. For example, Gini's mean difference is asymptotically efficient for a continuous distribution which is symmetric with respect to a point on R Its limiting risk deficiency is about 1.12 for a normal distribution.