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

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.