A sparse $p_0$ model with covariates for directed networks

TL;DR

This paper develops maximum likelihood estimation for a sparse directed network model with covariates, deriving error bounds and asymptotic normality results, supported by simulations and real data analysis.

Contribution

It provides the first comprehensive analysis of unrestricted MLE in a sparse $p_0$ model with covariates, including error bounds and asymptotic normality.

Findings

MLE errors are $O_p( (rac{ ext{log} n}{n})^{1/2} )$ for node parameters.

MLE errors are $O_p( rac{ ext{log} n}{n} )$ for regression coefficients.

Numerical studies confirm theoretical error bounds and normality.

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 $\bs{\eta}$ 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.