摘要：The $p_1$ exponential-family distribution in Holland and Leinhardt (1981) is one of the earliest models for directed random graphs and has been widely used in practice. The conditions for the existence and uniqueness of the maximum likelihood estimate (MLE) have been derived. However, it is a daunting task to investigate the large-sample properties of the MLE theoretically as the number of graphical vertices goes to infinity. The uniform consistency and asymptotic normality of the MLE have been derived for some specialized models closely related to the $p_1$ model but general results are lacking. In this article, we explore the consistency and asymptotic normality of the MLE in the $p_1$ model as the network size increases through numerical simulations. The results indicate that the $p_1$ model also enjoys good asymptotic properties.
关键词：directed random graphs; increasing number of parameters; maximum likelihood estimation; $p_1$ exponential-family distribution
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