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We are concerned here with unrestricted maximum likelihood estimation in asparse $p_0$ model with covariates for directed networks. The model has adensity parameter $\nu$, a $2n$-dimensional node parameter $\bs{\eta}$ and afixed dimensional regression coefficient $\bs{\gamma}$ of covariates. Previousstudies focus on the restricted likelihood inference. When the number of nodes$n$ goes to infinity, we derive the $\ell_\infty$-error between the maximumlikelihood estimator (MLE) $(\widehat{\bs{\eta}}, \widehat{\bs{\gamma}})$ andits 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 $\widehat{\bs{\gamma}}$, upto an additional factor. This explains the asymptotic bias phenomenon in theasymptotic normality of $\widehat{\bs{\gamma}}$ in\cite{Yan-Jiang-Fienberg-Leng2018}. Further, we derive the asymptotic normalityof the MLE. Numerical studies and a data analysis demonstrate our theoreticalfindings.

A sparse $p_0$ model with covariates for directed networks