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

We study a sparse negative binomial regression (NBR) for count data by showing the non-asymptotic advantages of using elastic-net estimator. Two types oracle inequalities are derived NBR's estimates Compatibility Factor Condition and Stabil Condition. The second type inequality is random design can be extended to many $\ell_1 + \ell_2$ regularized M-estimations, with corresponding empirical process having stochastic Lipschitz properties. derive concentration suprema processes weighted sum variables show some high--probability events. apply method sign consistency, provided that nonzero components in true vector larger than proper choice weakest signal detection threshold. In application, we grouping effect high probability. Third, under assumptions matrix, recover variable set probability if threshold large turning parameter up known constant. Lastly, briefly discuss de-biased estimator, numerical studies given support proposal.