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Asymptotics in directed exponential random graph models with an increasing bi-degree sequence
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.Comment: Published at http://dx.doi.org/10.1214/15-AOS1343 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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