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This paper studies the non-asymptotic merits of the double$\ell_1$-regularized for heterogeneous overdispersed count data via negativebinomial regressions. Under the restricted eigenvalue conditions, we prove theoracle inequalities for Lasso estimators of two partial regression coefficientsfor the first time, using concentration inequalities of empirical processes.Furthermore, derived from the oracle inequalities, the consistency andconvergence rate for the estimators are the theoretical guarantees for furtherstatistical inference. Finally, both simulations and a real data analysisdemonstrate that the new methods are effective.

Heterogeneous Overdispersed Count Data Regressions via Double Penalized Estimations