Parameter estimation via M- and Z-estimation is equally powerful in semiparametric models for one-dimensional functionals due to a one-to-one relation between corresponding loss and identification functions via integration and differentiation. For multivariate functionals such as multiple moments, quantiles, or the pair (Value at Risk, Expected Shortfall), this one-to-one relation fails and not every identification function possesses an antiderivative. The most important implication is an efficiency gap: The most efficient Z-estimator often outperforms the most efficient M-estimator. We theoretically establish this phenomenon for pairs of quantiles at different levels and for (Value at Risk, Expected Shortfall), and illustrate the gap numerically.

翻译：通过 M 和 Z 估计的参数估计在一维功能的半参数模型中同样有力,因为相应的损失和通过集成和区分的识别功能之间存在一对一的关系。对于多变功能,如多个时段、四分位数或对子(风险价值、预期缺省),这种一对一关系失败,而不是每个识别功能都具有抗降解作用。最重要的影响是效率差距:最有效的Z- 估计符往往优于效率最高的M- 估计符数。我们理论上为不同级别和不同级别对等的量子体和(风险价值、预期缺省)确定了这种现象,并用数字来说明差距。