Consistency of ${\mathrm{\ell }}_1$ Penalized Negative Binomial Regressions

We prove the consistency of the ${\mathrm{\ell }}_1$ penalized negative binomial regression (NBR). A real data application about German health care demand shows that the ${\mathrm{\ell }}_1$ penalized NBR produces a more concise but more accurate model, comparing to the classical NBR.