Statistical Inference For Partially Linear Varying Coefficient Model

With the expand of Internet database, the collected data volume is often high-dimensional in real problem. In order to deal with the problem, many parameter and semiparametric models are proposed to avoid "the curse of dimensionality". Among these models, the partially linear varying coefficient model contains linear and nonlinear part, not only contains a constant coefficient, but also contains the function coefficient. So it received widespread attention. The commonly used methods are the least squares estimation method (LSE) and the minimum variance method (MAVE) for the model,but the estimator we obtain may not is an efficient estimator though above methods. It requires us develop more suitable estimation method. On the other hand, because of the consistent accumulation of data, the number of covariates may increase at polynomial rate, and sometimes increase at an exponential rate. How to obtain better estimation and statistical inference for the parametric or semiparametric model, it is more and more important problem in the high dimensional data. As a result, when we do research on the problems of the partially linear varying coefficient model with the high-dimension and ultra-high dimension, we need develop a more brillant method. This paper systematically studies the estimation and variable selection for partially linear varying coefficient model in high-dimensional data. The results show that we can obtain effective estimation through the effective estimation equation; we can identify constant coefficient and functions variables using the group lasso method; we can reduce the dimension of high dimensional data though sorting KL distance for feature selection.This article focuses on the problems of the partially linear varying coefficient model with different dimension. The main contents are as follows:(1) The paper studies the efficiency problems of estimation in the partially linear varying coefficient model with heteroscedasticity. Efficient score vector function and efficient estimation of interest parameters are given under complete sample. Then we can construct the efficient estimating equation and give semiparametric efficient bound in the partially linear varying coefficient model with heteroscedasticity. We prove the solution of this equation is the efficient estimator for interest parameter, and the large sample properties for the efficient estimator are studied. Some numerical simulations are also proposed to investigate its finite sample properties.(2) Under the condition of high-dimension, the paper research the problem of variable selection of the partially linear varying coefficient model and put forward the two-stage variable selection method and do variable selection on the linear and nonlinear part of the model separately, achieving the parameter’s Adaptive Lasso estimation, proving the asymptotic and consistency properties of estimation, and research the finite sample properties of estimator through numerical simulation.(3)Under the condition of ultra-high dimensions, we use KL distance method to do feature screening in the ultra-high dimension sparse variable coefficient model. We construct the covariates between response with marginal KL distance statistic through conditional cumulative distribution function. Then we can screen variables in sort. The paper use numerical simulation to validate the finite sample property of the method proposed in this paper.