## The Annals of Statistics

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

#### 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.

#### Article information

Source
Ann. Statist., Volume 44, Number 1 (2016), 31-57.

Dates
Revised: May 2015
First available in Project Euclid: 10 December 2015

https://projecteuclid.org/euclid.aos/1449755956

Digital Object Identifier
doi:10.1214/15-AOS1343

Mathematical Reviews number (MathSciNet)
MR3449761

Zentralblatt MATH identifier
1331.62110

#### Citation

Yan, Ting; Leng, Chenlei; Zhu, Ji. Asymptotics in directed exponential random graph models with an increasing bi-degree sequence. Ann. Statist. 44 (2016), no. 1, 31--57. doi:10.1214/15-AOS1343. https://projecteuclid.org/euclid.aos/1449755956

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#### Supplemental materials

• Supplement to “Asymptotics in directed exponential random graph models with an increasing bi-degree sequence.”. The supplemental material contains proofs for the lemmas in Section 2.2, the theorems and lemmas in Sections 2.3 and 2.4, Proposition 1 and Theorem 7.