摘要翻译：
在高维稀疏模型中，我们考虑中值回归和更一般的分位数回归的可能无限集合。在这些模型中，回归子的总数$P$非常大，可能比样本量$N$还要大，但这些回归子中只有$S$对响应变量的条件分位数有非零影响，其中$S$的增长速度比$N$慢。我们考虑了系数的$\ell_1$-范数($\ell_1$-qr)对分位数回归的影响。首先，我们证明$\ell_1$-qr以$\sqrt{S/N}\sqrt{\log p}$的速率是一致的。回归子的总数$P$仅通过$\log P$因子影响汇率，因此零影响回归子的数量几乎呈指数增长。速率结果在相对较弱的条件下成立，要求$S/N$以超对数速度收敛到零，且正则化参数满足一定的理论约束。其次，我们提出了一个关键的，数据驱动的正则化参数的选择，并证明它满足这些理论约束。第三，当真模型的非零系数与零分离时，我们证明$\ell_1$-qr正确地选择真极小模型作为有效子模型。我们还证明了$\ell_1$-qr中非零系数的个数与$S$具有相同的随机顺序。第四，我们分析了将普通分位数回归应用于所选模型的两步估计的收敛速度。第五，我们在蒙特卡罗实验中评估了$\ell_1$-QR的性能，并说明了它在一个国际经济增长应用中的应用。
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英文标题：
《L1-Penalized Quantile Regression in High-Dimensional Sparse Models》
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作者：
Alexandre Belloni and Victor Chernozhukov
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最新提交年份：
2019
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Economics        经济学
二级分类：Econometrics        计量经济学
分类描述：Econometric Theory, Micro-Econometrics, Macro-Econometrics, Empirical Content of Economic Relations discovered via New Methods, Methodological Aspects of the Application of Statistical Inference to Economic Data.
计量经济学理论，微观计量经济学，宏观计量经济学，通过新方法发现的经济关系的实证内容，统计推论应用于经济数据的方法论方面。
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Statistics        统计学
二级分类：Methodology        方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
  We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models. In these models the overall number of regressors $p$ is very large, possibly larger than the sample size $n$, but only $s$ of these regressors have non-zero impact on the conditional quantile of the response variable, where $s$ grows slower than $n$. We consider quantile regression penalized by the $\ell_1$-norm of coefficients ($\ell_1$-QR). First, we show that $\ell_1$-QR is consistent at the rate $\sqrt{s/n} \sqrt{\log p}$. The overall number of regressors $p$ affects the rate only through the $\log p$ factor, thus allowing nearly exponential growth in the number of zero-impact regressors. The rate result holds under relatively weak conditions, requiring that $s/n$ converges to zero at a super-logarithmic speed and that regularization parameter satisfies certain theoretical constraints. Second, we propose a pivotal, data-driven choice of the regularization parameter and show that it satisfies these theoretical constraints. Third, we show that $\ell_1$-QR correctly selects the true minimal model as a valid submodel, when the non-zero coefficients of the true model are well separated from zero. We also show that the number of non-zero coefficients in $\ell_1$-QR is of same stochastic order as $s$. Fourth, we analyze the rate of convergence of a two-step estimator that applies ordinary quantile regression to the selected model. Fifth, we evaluate the performance of $\ell_1$-QR in a Monte-Carlo experiment, and illustrate its use on an international economic growth application.
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PDF链接：
https://arxiv.org/pdf/0904.2931