In this article, a weighted empirical likelihood technique for constructing the empirical likelihood confidence regions is applied to study the heteroscedastic varying coefficient partially non-linear models with missing response data. We first provide an estimator of the error variance based on the Nadaraya–Watson kernel estimation method. Then a weighted empirical log-likelihood ratio of the unknown parameter is constructed based on the inverse probability weighted technique. The maximum empirical likelihood (MEL) estimator of the unknown parameter is obtained. Further, a weighted empirical log-likelihood ratio of the varying coefficient function is introduced based on the MEL estimator and the inverse probability weighted method. The limiting distributions of the resulting statistics for both the unknown parameter and varying coefficient function are shown to have the standard chi-squared distribution. A simulation study and a real data set example are undertaken to investigate the finite sample performance of the proposed methods.

中文翻译：

---

缺失数据的异方差变系数部分非线性模型的加权经验似然

本文采用加权经验似然技术构建经验似然置信区域，研究缺少响应数据的异方差变系数部分非线性模型。我们首先基于Nadaraya–Watson核估计方法提供误差方差的估计器。然后，基于逆概率加权技术，构造了未知参数的加权经验对数似然比。获得未知参数的最大经验似然（MEL）估计量。此外，基于MEL估计器和逆概率加权方法，介绍了变系数函数的加权经验对数似然比。对于未知参数和变化系数函数，所得统计量的极限分布都显示为具有标准卡方分布。进行了仿真研究和实际数据集示例，以研究所提出方法的有限样本性能。