We study a sparse negative binomial regression (NBR) for count data by showing the non-asymptotic advantages of using the elastic-net estimator. Two types of oracle inequalities are derived for the NBR's elastic-net estimates by using the Compatibility Factor Condition and the Stabil Condition. The second type of oracle inequality is for the random design and can be extended to many $\ell_1 + \ell_2$ regularized M-estimations, with the corresponding empirical process having stochastic Lipschitz properties. We derive the concentration inequality for the suprema empirical processes for the weighted sum of negative binomial variables to show some high--probability events. We apply the method by showing the sign consistency, provided that the nonzero components in the true sparse vector are larger than a proper choice of the weakest signal detection threshold. In the second application, we show the grouping effect inequality with high probability. Third, under some assumptions for a design matrix, we can recover the true variable set with a high probability if the weakest signal detection threshold is large than the turning parameter up to a known constant. Lastly, we briefly discuss the de-biased elastic-net estimator, and numerical studies are given to support the proposal.
翻译:我们通过显示使用弹性网天文估计仪表天文图的不亚于自然的优势,对计数数据进行稀薄的负二进制回归(NBR)研究。我们通过使用兼容因子条件和稳定状态,为NBR的弹性网估算得出两种类甲不平等。第二种甲骨文不平等是随机设计的,可以扩大到许多美圆1+\ell_2美元,并具有相应的实验性进程具有随机利普希茨特性。我们为负二进制变量加权总和的超模实验性进程得出两种类甲骨骼不平等,以显示一些高概率事件。我们采用这种方法,通过显示符号一致性,前提是真正稀散矢量矢量的非零成分大于最弱信号检测阈值的适当选择。在第二个应用中,我们显示了高概率的分组效应不平等。第三,根据设计矩阵的一些假设,我们可以恢复真实变量的设定值,如果最弱的信号检测值为高概率,我们所认识的最小信号检测值将最终的参数变成一个不变的参数。