Variance Estimation for High-Dimensional Varying Index Coefficient Models

This paper studies the re-adjusted cross-validation method and a semiparametric regression model called the varying index coefficient model. We use the profile spline modal estimator method to estimate the coefficients of the parameter part of the Varying Index Coefficient Model (VICM), while the unknown function part uses the B-spline to expand. Moreover, we combine the above two estimation methods under the assumption of high-dimensional data. The results of data simulation and empirical analysis show that for the varying index coefficient model, the re-adjusted cross-validation method is better in terms of accuracy and stability than traditional methods based on ordinary least squares.

RCV-based error density estimation in the ultrahigh dimensional additive model

In this paper,we mainly study how to estimate the error density in the ultrahigh dimensional sparse additive model,where the number of variables is larger than the sample size.First,a smoothing method based on B-splines is applied to the estimation of regression functions.Second,an improved two-stage refitted crossvalidation(RCV)procedure by random splitting technique is used to obtain the residuals of the model,and then the residual-based kernel method is applied to estimate the error density function.Under suitable sparse conditions,the large sample properties of the estimator,including the weak and strong consistency,as well as normality and the law of the iterated logarithm,are obtained.Especially,the relationship between the sparsity and the convergence rate of the kernel density estimator is given.The methodology is illustrated by simulations and a real data example,which suggests that the proposed method performs well.

Robust Error Density Estimation in Ultrahigh Dimensional Sparse Linear Model

This paper focuses on error density estimation in ultrahigh dimensional sparse linear model,where the error term may have a heavy-tailed distribution.First,an improved two-stage refitted crossvalidation method combined with some robust variable screening procedures such as RRCS and variable selection methods such as LAD-SCAD is used to obtain the submodel,and then the residual-based kernel density method is applied to estimate the error density through LAD regression.Under given conditions,the large sample properties of the estimator are also established.Especially,we explicitly give the relationship between the sparsity and the convergence rate of the kernel density estimator.The simulation results show that the proposed error density estimator has a good performance.A real data example is presented to illustrate our methods.