A sparse $p_0$ model with covariates for directed networks

We are concerned here with unrestricted maximum likelihood estimation in a sparse $p_0$ model with covariates for directed networks. The model has a density parameter $ u$, a $2n$-dimensional node parameter $bs{eta}$ and a fixed dimensional regression coefficient $bs{gamma}$ of covariates. Previous studies focus on the restricted likelihood inference. When the number of nodes $n$ goes to infinity, we derive the $ell_infty$-error between the maximum likelihood estimator (MLE) $(widehat{bs{eta}}, widehat{bs{gamma}})$ and its true value $(bs{eta}, bs{gamma})$. They are $O_p( (log n/n)^{1/2} )$ for $widehat{bs{eta}}$ and $O_p( log n/n)$ for $widehat{bs{gamma}}$, up to an additional factor. This explains the asymptotic bias phenomenon in the asymptotic normality of $widehat{bs{gamma}}$ in cite{Yan-Jiang-Fienberg-Leng2018}. Further, we derive the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.