Summary

This article aims to shed light on inference problems for statistical models under alternative or nonstandard asymptotic frameworks from the perspective of the jackknife empirical likelihood. Examples include small-bandwidth asymptotics for semiparametric inference and goodness-of-fit testing, sparse-network asymptotics, many-covariates asymptotics for regression models, and many-weak-instruments asymptotics for instrumental variable regression. We first establish Wilks’ theorem for the jackknife empirical likelihood statistic in a general semiparametric inference problem under the conventional asymptotics. We then show that the jackknife empirical likelihood statistic may lose asymptotic pivotalness in the above nonstandard asymptotic frameworks, and argue that this phenomenon can be understood in terms of the emergence of Efron & Stein (1981)’s bias of the jackknife variance estimator at first order. Finally, we propose a modification of the jackknife empirical likelihood to recover asymptotic pivotalness under both conventional and nonstandard asymptotics. Our modification works for all of the above examples and provides a unified framework for investigating nonstandard asymptotic problems.

中文翻译：

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Jackknife 经验似然：小带宽、稀疏网络和高维渐近

概括

本文旨在从 jackknife 经验似然的角度阐明替代或非标准渐近框架下统计模型的推理问题。示例包括用于半参数推理和拟合优度检验的小带宽渐近，稀疏网络渐近，用于回归模型的多协变量渐近，以及用于工具变量回归的多弱工具渐近。我们首先在常规渐近法下的一般半参数推理问题中为 jackknife 经验似然统计量建立 Wilks 定理。然后我们证明了 jackknife 经验似然统计量可能会在上述非标准渐近框架中失去渐近关键性，并认为这种现象可以从 Efron & Stein (1981) 的一阶折刀方差估计量的偏差。最后，我们建议修改折刀经验可能性，以在常规和非标准渐近下恢复渐近关键性。我们的修改适用于上述所有示例，并为研究非标准渐近问题提供了统一的框架。