高维部分线性可加稳健Expectile回归模型

Partially Linear Additive Robust Expectile Regression in High Dimension

摘要: 高维数据一般因具有异方差或非齐次协变量而具有异质性，分位数回归和expectile回归是分析异质高维数据的有力工具，但前者由于损失函数非光滑的特性在计算方面存在较大挑战，而后者会因异常值而不稳健。本文利用一类稳健的非对称损失函数来研究部分线性可加模型的稳健expectile回归，用B样条基函数近似非参数部分，利用加入非凸惩罚的正则化方法来实现变量筛选并进行参数估计。该方法的优势在于：(1) 通过取不同分位水平得到响应变量更完整的条件分布，从而探索数据的异质性分布；(2) 部分线性的模型结构兼顾了线性解释变量和非线性解释变量，一方面增加了模型的灵活性，同时也具有一定的模型可解释性；(3) 稳健expectile回归估计比分位数回归方法计算效率高，比expectile回归稳健。数值模拟和实际数据分析均显示了该方法在模型估计和计算效率上的优势。

Abstract: High-dimensional data are generally heterogeneous due to heteroskedasticity or non-homogeneous covariates. Quantile regression and expectile regression are powerful tools for analyzing heterogeneous high-dimensional data, but the former is a great challenge in calculation due to the non-smooth nature of the loss function, while the latter is unstable due to outliers. In this paper, a class of robust asymmetric loss functions is used to study the robust expectile regression of partial linear additive models, the B-spline basis function is used to approximate the non-parametric part, and the regularization method with non-convex penalty is used to realize variable screening and parameter estimation. The advantages of this method are: (1) A more complete conditional distribution of response variables can be obtained by taking different quantile levels, so as to explore the heterogeneity distribution of data; (2) The partial linear model structure takes into account both linear explanatory variables and nonlinear explanatory variables, which increases the flexibility of the model on the one hand, and has a certain interpretability of the model; (3) The robust expectile regression estimation score digit regression method has higher computational efficiency and is more robust than the expectile regression. Both numerical simulation and actual data analysis show the advantages of the proposed method in model estimation and computational efficiency.