Error Variance Estimation In High Dimensional Varying Coefficient Models

In recent years,with the progress of science and technology,the demand of the data processing is becoming more and more important in genomics,finance,human health,image data and so on,Many scholars have shifted their focus to high dimensional data and even ultra-high dimensional data.The research on statistical inference based on high dimensional data is also developing,Statistical inference mainly solves four problems,including the variance estimation of random error,hypothesis testing of regression function,construction of confidence interval and variable selection.and the latter three aspects need to be based on the accurate estimation of the error variance.Therefore,it is significant to study the variance estimation of random errors in the high dimensional case.In this paper,error variance estimation is based on the general varying coefficient models,alas the varying of every coefficient is relying on the same latent variable.Varying coefficient models is an important category of non-parametric statistical models,which reflecting the varying trend of the model coefficients under the influence of the latent variable.Comparing with the general linear model,varying coefficient models have better explanation and prediction abilities,so this paper studies the error variance estimation in high dimensional varying coefficient model.This paper mainly mainly includes the following four aspects:Firstly,for the high dimensional varying coefficient models.In the premise of sparse assumption,this paper through constructing b-spline base to approximate the model coefficients,then model will turn into the general linear model;Next,then based on the correlation learning,this paper applies non-parametric independence screening method to screen important features,that is,alas ranking the marginal utility of every co-variant to achieve feature screening.which can realize the variable selection and improve the algorithm stability and accuracy of statistics;Then,this paper will split the data,apply refitted cross validation(RCV)method to implement the variance estimation of random error,which possess asymptotic normality in theory,and has good properties;Finally,This paper will do some numerical simulations on RCV estimation,naive estimation and oracle estimation,and oracle estimator is the unbiased estimation of real variance,and find that Naive estimates underestimate the true parameters,and the RCV estimator is very closed to oracle estimator.It turns out to be the case that under some wild conditions,applying RCV estimation method to varying coefficient models not only avoid the estimation hindering brought by the "curse of dimension",but also obtain an stable estimator,which has a better effect.