The class of weighted M-estimators is defined. The ratio of the asymptotic variance of the weighted estimator to the asymptotic variance of the optimally weighted estimator is defined as the inefficiency. A Kantorovich inequality is proved, its implications are investigated for the misweighted mean and misweighted median, and the results are applied to a batch of demographic data.

中文翻译：

---

存在不等标度时的 M 估计

定义了加权 M 估计量的类别。加权估计量的渐近方差与最优加权估计量的渐近方差之比被定义为无效率。证明了 Kantorovich 不等式，研究了其对加权均值和中位数的影响，并将结果应用于一批人口统计数据。