LetTjbe a reasonable estimator (for example, a minimum mean square error estimator) of the parameter θ of the familyDjof distributions,j= 1, 2, …,m.An estimatorT, which is a weighted mean ofT1,Ts, …,Tm, is found that has the same asymptotic distribution as that ofTj, when the sample comes fromDj,j= 1, 2, …,m.Here the weights are functions of the sample items. Empirical evidence is given which indicates thatTis satisfactory for small sample sizes. It is proved that ifTjand the weightWjare odd location and even location-free statistics, respectively,j= 1, 2, …,m, thenT= ΣWiTi, where ΣWi= 1, is an unbiased estimator of the center of every symmetric distribution, provided certain expectations exist. This is useful in the construction of the weight functionWj.