Empirical likelihood and Wilks phenomenon for data with nonignorable missing values

Abstract

Wilks's theorem is useful for constructing confidence regions. When applying the popular empirical likelihood to data with nonignorable nonresponses, Wilks's phenomenon does not hold. This paper unveils that this is caused by the extra estimation of the nuisance parameter in the nonignorable nonresponse propensity. Motivated by this result, we propose an adjusted empirical likelihood for which Wilks's theorem holds. Asymptotic results are presented and supplemented by simulation results for finite sample performance of the point estimators and confidence regions. An analysis of a data set is included for illustration.

Sampling Importance Resampling Algorithm with Nonignorable Missing Response Variable Based on Smoothed Quantile Regression

“…Furthermore, it is worth noting that if β and γ are known, we still have T l = A l , and in this case, the Wilks' Theorem still holds. The above conclusion is consistent with Zhao [26].…”

Quantile regression and smoothed empirical likelihood for non-ignorable missing data based on semi-parametric response models

“…注意到条件 (A1)-(A5) 与 Whang [26] 的假设 1-6 类似, 用于说明分位回归系数平滑经验似然估计 的渐近正态性质并导出经验似然比统计量的 Taylor 展开式. 此外, 估计量 θ(l) (l = 1, 2, 3) 的渐近性质 还依赖于倾斜参数估计量 γ 的渐近性质, 不可忽略缺失数据分析相关文献的常见条件 (A6)-(A9) 为此 提供了保证 [34] . 在假设条件满足的情形下, 定义…”

Semiparametric recovery of central dimension reduction space with nonignorable nonresponse

“…On the other hand, follow‐up studies can be avoided if there exists an instrumental variable for the nonresponse. Many studies have adopted the instrumental variable approach and showed that it works well; see, for example, Shao and Wang (2016), Wang et al (2014), Zhang et al (2018), Zhao, Wang, and Shao (2019).…”

“…In this case, it is generally thought that given the study variable $$$ Y $$$ and the measured covariates, the pretreatment baseline covariates are independent of the propensity score, but clearly, they are correlated with $$$ Y $$$. An example can be found in Zhao et al (2019). The instrumental variable can often be constructed from the data.…”

“…Remark Unlike theorem 4.1 of Zhao et al (2021) that uses multiple dimension reductions on $$$ \mathbf{X} $$$ to construct a joint dimension reduction space, our method requires only a single implementation of dimension reduction on the covariate vector $$$ \mathbf{U} $$$, which is the $$$ \mathbf{X} $$$ vector excluding the instrumental variable. Our method has a computational advantage over existing methods and can serve as an analytical basis for methods that involve parameter estimation with known instrumental variables under the nonignorable data missingness, for exampleShao and Wang (2016), Zhang et al (2018), and Zhao et al (2019). …”