**影响因子：**0.0

**DOI码：**10.27405/d.cnki.gxbdu.2022.002063

**发表刊物：**Chinese Journal of Applied Probability and Statistics

**关键字：**Partially linear models; Semi-linear neural networks; Estimator; Consistency; Gradient descent

**摘要：**Neural networks can be applied as an estimation method for statistical models, which has been used to solve the estimation problem of nonparametric regression models successfully. However, the estimation method based on the neural network for the partially linear model has not been explored systematically.
For the estimation method of non-parametric regression model based on traditional neural networks, there exist two main shortcomings, which are the poor interpretability and the limited capabilities to generalize the global trends and local changes simultaneously, lead to the problem in the estimation of regression function in partially linearity model directly; Besides, although the neural network has been applied in various areas, it is necessary to be studied theoretical, so it is important to dig out the mathematical statistics properties behind the neural networks from a statistical perspective. To solve above issues, we present an estimation method for partially linear model based on neural
network, including the following two parts, to be specific:
In the first part, in response to the above problems, partially linear model estimation method based on a kind of semi-linear neural networks with single hidden layer is proposed. The strong approximation ability of this network and the consistency of the estimator based on this method are proved under some necessary conditions. Subsequently, a local back-propagation algorithm based on the idea of gradient descent is designed for approximate calculation of estimator parameters which matches with the single hidden layer semi-linear network structure. In addition, the property of the large samples is verified by numerical simulations, and the case analysis based on the Boston Housing Price data set confirms the necessity of setting the linear part in the traditional network with single hidden layer to solve the partially linear model estimation problem.
In the second part, considering the scalability of the new method, the partially linear model estimation method based on the semi-linear deep neural networks is further studied. The universal approximation theorem and the consistency theorem of estimator are proposed and proved under some necessary conditions, a local back-propagation algorithm that matches the semi-linear deep network structure is designed. The property of large amples is verified by numerical simulations, and the case analysis shows that it is still necessary to add linear parts in traditional deep neural networks.
Finally, the main contributions of this article is summarized and some questions that can be further studied in the future is
proposed.

**论文类型：**期刊论文

**ISSN号：**1001-4268

**是否译文：**否

**第一作者：**Zhiwei Liu

**通讯作者：**Zhiming Xia