Semi-Linear Neural Networks Estimation of Partially Linear Model

Based on the poor interpretability and the limitation of summarizing the overall trends and local changes at the same time of the traditional neural network, it is not suitable for estimating the regression function of the partial linear model directly. In response to this problem, the semi-linear neural network structure that has both linear and non-linear parts is constructed firstly. Then, the consistency of the network estimator based on empirical risk minimization is proved under some necessary conditions, and the semi-linear network parameter estimation algorithm based on gradient descent is designed, which is called as the local back propagation algorithm. The random simulation experiments verify the large sample property, the results of the case analysis explain the necessity of introducing a linear part in the neural network. In particular, the experiment shows that the estimation effect of this method is slightly better than the N-W kernel estimation method based on the Boston House Price Dataset.