In real applications of the linear model, the explanatory variables are very often naturally grouped, the most common example being the multivariate variance analysis. In the present paper, a quantile model with group structure is considered, where the number of groups can diverge with the sample size. We introduce and study the adaptive elastic-net group quantile estimator, for improving the parameter estimation accuracy. This method allows automatic selection, with a probability converging to 1, of the non-zero coefficient vectors and further, the asymptotic normality of the non-zero parameter estimators. We also give the convergence rate of the adaptive elastic-net group quantile estimator, rate which depends on the number of the groups. In order to put the estimation method into practice, an algorithm based on the subgradient method is proposed and implemented. The performed Monte Carlo simulations show that the adaptive elastic-net group quantile estimations are more accurate than other existing group estimations in the literature. Moreover, the numerical study confirms the theoretical results and the usefulness of the proposed estimation method.

中文翻译：

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分位数模型中具有不同变量组数的自适应弹性网选择

在线性模型的实际应用中，解释变量通常是自然分组的，最常见的例子是多元方差分析。在本文中，考虑了具有组结构的分位数模型，其中组的数量可以随着样本大小而发散。我们引入并研究了自适应弹性网群分位数估计器，以提高参数估计精度。该方法允许以收敛到 1 的概率自动选择非零系数向量以及非零参数估计量的渐近正态性。我们还给出了自适应弹性网组分位数估计器的收敛速度，收敛速度取决于组的数量。为了将估计方法付诸实践，提出并实现了一种基于次梯度法的算法。执行的蒙特卡罗模拟表明，自适应弹性网组分位数估计比文献中其他现有的组估计更准确。此外，数值研究证实了所提出的估计方法的理论结果和有用性。