Error density estimation in high-dimensional sparse linear model

This paper is concerned with the error density estimation in high-dimensional sparse linear model, where the number of variables may be larger than the sample size. An improved two-stage refitted cross-validation procedure by random splitting technique is used to obtain the residuals of the model, and then traditional kernel density method is applied to estimate the error density. Under suitable sparse conditions, the large sample properties of the estimator including the consistency and asymptotic normality, as well as the law of the iterated logarithm are obtained. Especially, we gave the relationship between the sparsity and the convergence rate of the kernel density estimator. The simulation results show that our error density estimator has a good performance. A real data example is presented to illustrate our methods.