A sparse $p_0$ model with covariates for directed networks

Abstract

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 $\nu$, a $2n$-dimensional node parameter $\bsη$ and a fixed dimensional regression coefficient $\bsγ$ 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) $(\widehat{\bsη}, \widehat{\bsγ})$ and its true value $(\bsη, \bsγ)$. They are $O_p((\log n/n{)}^{1/2})$ for $\widehat{\bsη}$ and $O_p(\log n/n)$ for $\widehat{\bsγ}$, up to an additional factor. This explains the asymptotic bias phenomenon in the asymptotic normality of $\widehat{\bsγ}$ 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.

19 pages,2 figures,3 tables. arXiv admin note: substantial text overlap with arXiv:1609.04558 by other authors

References in corpus (3)

• GEE analysis of clustered binary data with diverging number of covariates
• Asymptotics in directed exponential random graph models with an increasing bi-degree sequence
• Maximum entropy distributions on graphs