Jackknife empirical likelihood: small bandwidth, sparse network and high-dimensional asymptotics

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.