Efficient estimation and variable selection in partial linear varying coefficient quantile regression model with longitudinal data is concerned in this paper. To improve estimation efficiency in quantile regression, based on B-spline basis approximation for nonparametric parts, we propose a new estimating function, which can incorporate the correlation structure between repeated measures. In order to reduce computational burdens, the induced smoothing method is used. The new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations based methods. Under mild conditions, the asymptotically normal distribution of the estimators for the parametric components and the optimal convergence rate of the estimators for the nonparametric functions are established. Furthermore, to do variable selection, a smooth-threshold estimating equation is proposed, which can use the correlation structure and select the nonparametric and parametric parts simultaneously. Theoretically, the variable selection procedure works beautifully, including consistency in variable selection and oracle property in estimation. Simulation studies and real data analysis are included to show the finite sample performance.

中文翻译：

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具有纵向数据的偏线性变系数分位数回归模型中的有效参数估计和变量选择

本文研究了具有纵向数据的偏线性变系数分位数回归模型中的有效估计和变量选择。为了提高分位数回归中的估计效率，基于非参数部分的 B 样条基近似，我们提出了一种新的估计函数，该函数可以结合重复测量之间的相关结构。为了减少计算负担，使用了诱导平滑方法。经验表明，新方法比流行的基于广义估计方程的方法更有效、更稳健。在温和条件下，建立了参数分量估计量的渐近正态分布和非参数函数估计量的最优收敛速度。此外，要进行变量选择，提出了一个平滑阈值估计方程，该方程可以利用相关结构，同时选择非参数和参数部分。从理论上讲，变量选择程序运行良好，包括变量选择的一致性和估计中的预言机属性。包括模拟研究和真实数据分析以显示有限样本性能。