l1-PENALIZED QUANTILE REGRESSION IN HIGH-DIMENSIONAL SPARSE MODELS
l1-PENALIZED QUANTILE REGRESSION IN HIGH-DIMENSIONAL SPARSE MODELS
复制标题
DOI：
10.1214/10-aos827
复制
发表时间：
2011-02-01
影响因子：
4.5
通讯作者：
Chernozhukov, Victor
中科院分区：
文献类型：
--
作者：
Belloni, Alexandre;Chernozhukov, Victor
我们考虑中值回归，更普遍地是在高维稀疏模型中可能无限的分位数回归集合。在这些模型中，回归器p的数量非常大，可能比样本量N大，但是大多数S回归器仅对响应变量的每个条件分位数都有非零影响，其中S比N较慢。由于在这种情况下普通的分位数回归不一致，因此我们考虑l（1）二元分位数回归（L（1）-QR），该回归损失了回归系数的L（1） - 量子，以及后载后的。 QR估计器（l（1）-QR），将普通QR应用于L（1）-QR选择的模型。首先，我们表明在一般条件下l（1）-QR在近端速率下是一致的。 root s/n根log（p boolean或n），在分位数索引的（0，1）的紧凑型集合中均匀。在得出这一结果时，我们提出了部分关键的，数据驱动的罚款水平的选择，并表明它满足实现此速度的要求。其次，我们表明在类似条件下，l（1）-QR在近门速率root s/n root log（p boolean或n）上是一致的，即使l（1）-qr也均匀地在u上。 - 选择的模型错过了真实模型的某些组件，否则速率可能更接近Oracle速率。第三，我们表征了l（1）-QR包含真实模型作为子模型的条件，并在所选模型的维度上得出界限，均匀地在u上。我们还提供了条件，在这些条件下，势头限制选择最小的真实模型，均匀地超过u。
We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models. In these models, the number of regressors p is very large, possibly larger than the sample size n, but only at most s regressors have a nonzero impact on each conditional quantile of the response variable, where s grows more slowly than n. Since ordinary quantile regression is not consistent in this case, we consider l(1)-penalized quantile regression (l(1)-QR), which penalizes the l(1)-norm of regression coefficients, as well as the post-penalized QR estimator (post-l(1)-QR), which applies ordinary QR to the model selected by l(1)-QR. First, we show that under general conditions l(1)-QR is consistent at the near-oracle rate. root s/n root log(p boolean OR n), uniformly in the compact set u subset of (0, 1) of quantile indices. In deriving this result, we propose a partly pivotal, data-driven choice of the penalty level and show that it satisfies the requirements for achieving this rate. Second, we show that under similar conditions post-l(1)-QR is consistent at the near-oracle rate root s/n root log(p boolean OR n), uniformly over u, even if the l(1)-QR-selected models miss some components of the true models, and the rate could be even closer to the oracle rate otherwise. Third, we characterize conditions under which l(1)-QR contains the true model as a submodel, and derive bounds on the dimension of the selected model, uniformly over u; we also provide conditions under which hard-thresholding selects the minimal true model, uniformly over u.