一阶退化泛函的推论
Inference on Functionals under First Order Degeneracy
摘要：
本文提出了一个统一的二阶渐近框架进行推断的参数$ϕ(θ_0)$的形式，其中$θ_0$是未知的，但可以估计由$\hatθ_n$，$ϕ$是一个已知的地图，允许零一阶导数在$θ_0$。对于文献中的大量例子，二阶Delta方法揭示了插入式估计量$ϕ(\hatθ_n)$的非退化弱极限。然而，我们表明，“标准”引导是一致的，当且仅当二阶导数$ϕ_{θ_0}''=0$在正则性条件下，即，如果是$ϕ_{θ_0}''\neq 0$，则标准自举是不一致的，并且提供了对推断没有帮助的退化极限。因此，我们确定了与Fang和Santos（2018）不同的引导失败的来源，因为即使$ϕ$是可微的，问题（始终引导\textit{nondegenerate}极限）仍然存在。我们证明了Babu（1984）中的修正过程可以推广到我们的一般设置。另外，当映射是\textit{in addition}二阶不可微映射时，提出了一种改进的自举方法。两者都显示在某些条件下提供局部尺寸控制。作为一个例子，我们开发了一个测试的共同条件heteroskedastic（CH）功能，设置与退化和不可微-后者是因为雅可比矩阵是退化为零，我们允许存在多个共同的CH功能。
英文摘要：
This paper presents a unified second order asymptotic framework for conducting inference on parameters of the form $ϕ(θ_0)$, where $θ_0$ is unknown but can be estimated by $\hatθ_n$, and $ϕ$ is a known map that admits null first order derivative at $θ_0$. For a large number of examples in the literature, the second order Delta method reveals a nondegenerate weak limit for the plug-in estimator $ϕ(\hatθ_n)$. We show, however, that the `standard' bootstrap is consistent if and only if the second order derivative $ϕ_{θ_0}''=0$ under regularity conditions, i.e., the standard bootstrap is inconsistent if $ϕ_{θ_0}''\neq 0$, and provides degenerate limits unhelpful for inference otherwise. We thus identify a source of bootstrap failures distinct from that in Fang and Santos (2018) because the problem (of consistently bootstrapping a \textit{nondegenerate} limit) persists even if $ϕ$ is differentiable. We show that the correction procedure in Babu (1984) can be extended to our general setup. Alternatively, a modified bootstrap is proposed when the map is \textit{in addition} second order nondifferentiable. Both are shown to provide local size control under some conditions. As an illustration, we develop a test of common conditional heteroskedastic (CH) features, a setting with both degeneracy and nondifferentiability -- the latter is because the Jacobian matrix is degenerate at zero and we allow the existence of multiple common CH features.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
