We study sparse high-dimensional negative binomial regression problem for count data regression by showing non-asymptotic merits of the Elastic-net regularized estimator. With the KKT conditions, we derive two types of non-asymptotic oracle inequalities for the elastic net estimates of negative binomial regression by utilizing Compatibility factor and Stabil Condition, respectively. Based on oracle inequalities we proposed, we firstly show the sign consistency property of the Elastic-net estimators provided that the non-zero components in sparse true vector are large than a proper choice of the weakest signal detection threshold, and the second application is that we give an oracle inequality for bounding the grouping effect with high probability, thirdly, under some assumptions of design matrix, we can recover the true variable set with high probability if the weakest signal detection threshold is large than 3 times the value of turning parameter, at last, we briefly discuss the de-biased Elastic-net estimator.

中文翻译：

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Elastic-net 正则化高维负二项式回归：一致性和弱信号检测

我们通过展示 Elastic-net 正则化估计器的非渐近优点来研究计数数据回归的稀疏高维负二项式回归问题。在 KKT 条件下，我们分别利用兼容性因子和稳定条件为负二项式回归的弹性净估计导出了两种类型的非渐近预言不等式。根据我们提出的Oracle不等式，我们首先显示了弹性净估计的标志一致性，条件是，稀疏的真载量的非零分量大于最弱的信号检测阈值的正确选择，而第二个应用程序我们给出了一个预言不等式，用于以高概率限制分组效应，第三，在设计矩阵的一些假设下，