Elastic-net Regularized High-dimensional Negative Binomial Regression: Consistency and Weak Signals Detection

We study sparse negative binomial regression (NBR) for count data by showing non-asymptotic merits of the Elastic-net estimator. Two types of oracle inequalities are derived for the Elastic-net estimates of NBR by utilizing Compatibility Factor or Stabil Condition. The second-type oracle inequality is for random design which can be extended to many $\ell_1 + \ell_2$ regularized M-estimation with the corresponding empirical process having stochastic Lipschitz properties. To show some high probability events, we derive concentration inequality for suprema empirical processes for the weighted sum of negative binomial variables. For applications, we show the sign consistency provided that the non-zero components in sparse true vector are larger than a proper choice of the weakest signal detection threshold; and the second application is that we show the grouping effect inequality 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 the turning parameter up to a known constant; at last, we briefly discuss the de-biased Elastic-net estimator and numerical studies are given to support the proposal.