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英文文献：Asymptotically Honest Confidence Regions for High Dimensional Parameters by the Desparsified Conservative Lasso-用解压缩保守套索研究高维参数的渐近可信域
英文文献作者：Mehmet Caner,Anders Bredahl Kock
英文文献摘要：
While variable selection and oracle inequalities for the estimation and prediction error have received considerable attention in the literature on high-dimensional models, very little work has been done in the area of testing and construction of confidence bands in high-dimensional models. However, in a recent paper van de Geer et al. (2014) showed how the Lasso can be desparsified in order to create asymptotically honest (uniform) confidence band. In this paper we consider the conservative Lasso which penalizes more correctly than the Lasso and hence has a lower estimation error. In particular, we develop an oracle inequality for the conservative Lasso only assuming the existence of a certain number of moments. This is done by means of the Marcinkiewicz-Zygmund inequality which in our context provides sharper bounds than Nemirovski's inequality. As opposed to van de Geer et al. (2014) we allow for heteroskedastic non-subgaussian error terms and covariates. Next, we desparsify the conservative Lasso estimator and derive the asymptotic distribution of tests involving an increasing number of parameters. As a stepping stone towards this, we also provide a feasible uniformly consistent estimator of the asymptotic covariance matrix of an increasing number of parameters which is robust against conditional heteroskedasticity. To our knowledge we are the first to do so. Next, we show that our confidence bands are honest over sparse high-dimensional sub vectors of the parameter space and that they contract at the optimal rate. All our results are valid in high-dimensional models. Our simulations reveal that the desparsified conservative Lasso estimates the parameters much more precisely than the desparsified Lasso, has much better size properties and produces confidence bands with markedly superior coverage rates.
虽然在高维模型的文献中，关于估计和预测误差的变量选择和oracle不等式得到了相当多的关注，但在高维模型置信带的检验和构建方面却鲜有研究。然而，在最近的一篇论文中，van de Geer等人(2014)展示了如何通过脱parsified套索来创建渐近诚实(一致)置信带。在本文中，我们考虑了保守套索，它比传统的套索具有更准确的惩罚效果，因而具有更低的估计误差。特别地，我们开发了保守套索的oracle不等式，仅假设存在一定的矩数。这是通过Marcinkiewicz-Zygmund不等式来实现的，在我们的文章中，它提供了比内米洛夫斯基不等式更清晰的界限。与van de Geer等人(2014)相反，我们允许异方差非亚高斯误差项和协变量。然后，我们解压缩保守拉索估计量，并推导出涉及越来越多参数的试验的渐近分布。作为此问题的基础，我们还给出了一个参数渐近协方差矩阵的可行一致一致估计，该估计对条件异方差具有鲁棒性。据我们所知，我们是第一个这样做的。接下来，我们证明了我们的置信带在参数空间的稀疏高维子向量上是诚实的，并且它们以最优速率收缩。所有的结果在高维模型中都是有效的。模拟结果表明，去加密保守套索比去加密保守套索能更精确地估计参数，具有更好的尺寸特性，并能产生覆盖率明显优于传统套索的置信带。
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