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.

中文翻译：

---

高维稀疏线性模型中的误差密度估计

本文关注的是高维稀疏线性模型中的误差密度估计，其中变量数量可能大于样本量。通过随机分裂技术改进的两阶段重新拟合交叉验证程序用于获得模型的残差，然后应用传统的核密度方法来估计误差密度。在合适的稀疏条件下，得到了估计量的大样本性质，包括一致性和渐近正态性，以及迭代对数定律。特别是，我们给出了核密度估计器的稀疏性和收敛速度之间的关系。仿真结果表明，我们的误差密度估计器具有良好的性能。提供了一个真实的数据示例来说明我们的方法。