Parametric Modeling of Quantile Regression Coefficient Functions with Longitudinal Data

In ordinary quantile regression, quantiles of different order are estimated one at a time. An alternative approach, which is referred to as <i>quantile regression coefficients modeling</i> (qrcm), is to model quantile regression coefficients as parametric functions of the order of the quantile. In this paper, we describe how the QRCM paradigm can be applied to longitudinal data. We introduce a two-level quantile function, in which two different quantile regression models are used to describe the (conditional) distribution of the within-subject response and that of the individual effects. We propose a novel type of penalized fixed-effects estimator, and discuss its advantages over standard methods based on ℓ1 and ℓ2 penalization. We provide model identifiability conditions, derive asymptotic properties, describe goodness-of-fit measures and model selection criteria, present simulation results, and discuss an application. The proposed method has been implemented in the R package qrcm.

普通分位数回归中，不同阶数的分位数需逐个单独估计。另一种被称为分位数回归系数建模（quantile regression coefficients modeling，qrcm）的方法，则将分位数回归系数建模为分位数阶数的参数化函数。本文阐述了如何将QRCM范式应用于纵向数据。我们引入两水平分位数函数，该函数采用两种不同的分位数回归模型，分别刻画被试内响应的（条件）分布与个体效应的分布。我们提出了一种新型惩罚固定效应估计量，并讨论了其相较于基于L1和L2惩罚的标准方法的优势。本文给出了模型可识别性条件，推导了渐近性质，介绍了拟合优度指标与模型选择准则，展示了模拟结果，并讨论了一个实际应用案例。所提方法已在R包qrcm中实现。