Estimation of covariate-dependent conditional covariance matrix in a high-dimensional space poses a challenge to contemporary statistical research. The existing kernel estimators may not be locally adaptive due to using a single bandwidth to explore the smoothness of all entries of the target matrix function. In this paper, we propose a novel framework to address this issue, where we factorize the target matrix into factors and estimate these factors in turn by the kernel approach. The resulting estimator is further regularized by thresholding and optimal shrinkage. Under certain mixing and sparsity conditions, we show that the proposed estimator is well-conditioned and uniformly consistent with the underlying matrix function even when the sample is dependent. Simulation studies suggest that the proposed estimator significantly outperforms its competitors in terms of integrated root-squared estimation error. We present an application to financial return data.

中文翻译：

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高维非参数协方差模型的分解估计

在高维空间中估计依赖于协变量的条件协方差矩阵对当代统计研究提出了挑战。由于使用单个带宽来探索目标矩阵函数的所有条目的平滑度，现有的内核估计器可能不是局部自适应的。在本文中，我们提出了一个新的框架来解决这个问题，我们将目标矩阵分解为因子，并通过核方法依次估计这些因子。通过阈值化和最优收缩进一步规范化得到的估计量。在某些混合和稀疏条件下，我们表明即使在样本相关的情况下，所提出的估计器也具有良好的条件并且与基础矩阵函数一致。模拟研究表明，所提出的估计器在综合平方根估计误差方面明显优于其竞争对手。我们提出了一个财务回报数据的应用程序。