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Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

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Yan, Ting, Leng, Chenlei and Zhu, Ji (2016) Asymptotics in directed exponential random graph models with an increasing bi-degree sequence. The Annals of Statistics, 44 (1). pp. 31-57. doi:10.1214/15-AOS1343

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Official URL: http://dx.doi.org/10.1214/15-AOS1343

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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 the 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 and non-binary weighted edges. We establish the uniform consistency and the asymptotic normality of the maximum likelihood estimator, 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. Along the way, we also establish a geometrically fast rate of convergence for the Newton iterative algorithm, which is used to obtain the maximum likelihood estimate. Numerical studies confirm our theoretical findings.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Mathematical statistics -- Graphic methods -- Research
Journal or Publication Title: The Annals of Statistics
Publisher: Institute of Mathematical Statistics
ISSN: 0090-5364
Official Date: 2016
Dates:
DateEvent
2016Published
10 December 2015Available
12 September 2015Accepted
Volume: 44
Number: 1
Page Range: pp. 31-57
DOI: 10.1214/15-AOS1343
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: Guo jia zi ran ke xue ji jin wei yuan hui (China) [National Natural Science Foundation of China] (NSFC), National Science Foundation (U.S.) (NSF)
Grant number: No. 11401239 (NSFC), DMS-14-07698 (NSF)

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