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Quantile function regression and variable selection for sparse models
The Canadian Journal of Statistics ( IF 0.6 ) Pub Date : 2021-04-23 , DOI: 10.1002/cjs.11616
Takuma Yoshida

This article considers linear quantile regression and variable selection for high-dimensional data. In general, an ordinary quantile regression estimator is obtained for a single, fixed quantile level. Therefore, the estimated coefficient does not have continuity with respect to the quantile level, and hence, the behaviour of the estimator and estimated active variable set could change rapidly for different but sufficiently close quantile levels. To obtain a stable estimator for a given quantile level, this study proposes a new quantile regression method to estimate the coefficient as a function of the quantile level of interest in a given region Δ ( 0 , 1 ), which is denoted quantile function regression. In quantile function regression, we approximate the coefficient function of the quantile level using a B-spline model, and hence, the estimated conditional quantile is continuous as it is a B-spline curve. To employ variable selection, a group lasso-type sparse penalty is used to estimate a non-zero coefficient function of the quantile level, which indicates the estimated active set that remains unchanged in Δ. Therefore, quantile function regression can achieve global variable selection. The proposed estimator exhibits an asymptotic rate of convergence and consistency in variable selection. Simulation studies and applications to real data further reveal that the proposed method yields good performance.

中文翻译:

稀疏模型的分位数函数回归和变量选择

本文考虑高维数据的线性分位数回归和变量选择。通常,对于单个固定的分位数水平,可以获得普通的分位数回归估计量。因此,估计的系数相对于分位数水平不具有连续性,因此,估计量和估计的活动变量集的行为可能会在不同但足够接近的分位数水平下迅速变化。为了获得给定分位数水平的稳定估计量,本研究提出了一种新的分位数回归方法来估计系数作为给定区域中感兴趣的分位数水平的函数 Δ ( 0 , 1 ),表示分位数函数回归。在分位数函数回归中,我们使用B样条模型来近似分位数水平的系数函数,因此,估计的条件分位数是连续的,因为它是B样条曲线。为了采用变量选择,使用组套索类型的稀疏惩罚来估计分位数水平的非零系数函数,这表明估计的活动集在 Δ. 因此,分位数函数回归可以实现全局变量选择。所提出的估计器在变量选择中表现出渐近的收敛速度和一致性。模拟研究和对真实数据的应用进一步表明,所提出的方法产生了良好的性能。
更新日期:2021-04-23
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