ASYMPTOTICS IN DIRECTED EXPONENTIAL RANDOM GRAPH MODELS WITH AN INCREASING BI-DEGREE SEQUENCE

Citation
Ting Yan et al., ASYMPTOTICS IN DIRECTED EXPONENTIAL RANDOM GRAPH MODELS WITH AN INCREASING BI-DEGREE SEQUENCE, Annals of statistics , 44(1), 2016, pp. 31-57
Journal title
ISSN journal
00905364
Volume
44
Issue
1
Year of publication
2016
Pages
31 - 57
Database
ACNP
SICI code
Abstract
Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.