Variance Estimation Of Single-index Model With High-dimensional Data

With the development of science and technology,all areas are generating a lot of data at all times,and the complexity of these data is getting higher and higher.When the dimension p is greater than the sample size n,it is said that the data is "high dimension" data.In this paper,we study the variance estimation of single-index model in high-dimensional case,On the one hand,variance estimation is a very important problem in statistical inference.On the other hand,the single-index model has the characteristics of flexibility and dimensionality,which makes it an effective model to deal with high-dimensional data.Therefore,the research of this paper is not only theoretically important but also has a wide range of uses in practice.For the single-index model,we propose RCV(Refitted Cross-Validation)method to estimate its variance,and theoretically proved to have the nature of Oracle.RCV method can improve the common two-step variance estimation(the first step Select the variable,the second step variance estimate).Meanwhile,we present the Monte Carlo simulation of the model based on the B-spline approximation.The variance estimators of the RCV-LASSO and RCV-SCAD methods are calculated and compared with the Oracle method,CV-LASSO and CV-SCAD.The results show that the variance,the deviation and the standard error of the RCV method are better than those of other methods.The variance estimator and the average model size of the RCV method are closer to the estimate of the Oracle model.RCV method greatly reduces the influence of the pseudo-correlation variable,which is more effective as the variable dimension increases.Finally,we use an actual data set to illustrate the effectiveness of theoretical results and numerical simulations.