Model structural inference on semiparametric measurement error models have not been well developed in the existing literature, partially due to the difficulties in dealing with unobservable covariates. In this study, a framework for adaptive structure selection is developed in partially linear error-in-function models with error-prone covariates. Firstly, based on the profile-least-square estimators of the current models, we define two test statistics via generalized likelihood ratio (GLR) test method (Fan et al. in Ann Stat 29(1):153–193, 2001). The proposed test statistics are shown to possess the Wilks-type properties, and a class of new Wilks phenomenon is unveiled in the family of semiparametric measurement error models. Then, we demonstrate that the GLR statistics asymptotically follow chi-squared distributions under null hypotheses. Further, we propose efficient algorithms to implement our methodology and assess the finite sample performance by simulated examples. A real example is given to illustrate the performance of the present methodology.

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具有易错协变量的部分线性函数错误模型的自适应结构推论

在现有文献中，关于半参数测量误差模型的模型结构推论尚未得到很好的发展，部分原因是难以处理不可观测的协变量。在这项研究中，在具有易错协变量的部分线性函数误差模型中，开发了一种自适应结构选择的框架。首先，基于当前模型的轮廓最小二乘估计，我们通过广义似然比（GLR）检验方法定义了两个检验统计量（Fan等人，Ann Stat 29（1）：153-193，2001）。建议的测试统计数据显示具有Wilks型属性，并且在半参数测量误差模型族中揭示了一类新的Wilks现象。然后，我们证明了在零假设下GLR统计量渐近地遵循卡方分布。进一步，我们提出了有效的算法来实施我们的方法并通过模拟示例评估有限的样本性能。给出一个真实的例子来说明本方法的性能。