When addressing semiparametric problems with parametric restrictions (assumptions on the distribution), the efficient score (ES) of a parameter is often important for generating useful estimates. However, usual derivation of ES, although conceptually simple, is often lengthy and with many steps that do not help in understanding why its final form arises. This drawback often casts onto semiparametric estimation a mantle that can turn away otherwise able doctoral students or researchers. Here we show that many ESs can be obtained as a one-step derivation after we characterize those features (envelopes) of the unrestricted problem that are constrained in the restricted problem. We demonstrate our arguments in three problems with known ES but whose usual derivations are lengthy. We show that the envelope-based derivation is dramatically explanatory and compact, needing essentially two lines where the standard approach needs 10 or more pages. This suggests that the envelope method can add useful intuition and exegesis to both teaching and research of semiparametric estimation.

中文翻译：

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参数约束下简明计算半参数有效分数的信封法

在解决具有参数限制（分布假设）的半参数问题时，参数的有效分数 (ES) 通常对于生成有用的估计很重要。然而，ES 的通常推导虽然在概念上很简单，但通常很冗长，并且有许多步骤无助于理解其最终形式出现的原因。这个缺点通常会给半参数估计带来一个可能会拒绝其他有能力的博士生或研究人员的地幔。在这里，我们展示了在我们表征了受限问题中受限的非受限问题的那些特征（包络）之后，可以通过一步推导获得许多 ES。我们在三个已知 ES 的问题中证明了我们的论点，但这些问题的推导通常很冗长。我们表明，基于信封的推导具有显着的解释性和紧凑性，基本上需要两行，而标准方法需要 10 页或更多页。这表明包络法可以为半参数估计的教学和研究增加有用的直觉和解释。