## The Annals of Statistics

### Maximum lilkelihood estimation in the $\beta$-model

#### Abstract

We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the $\beta$-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases.

#### Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1085-1110.

Dates
First available in Project Euclid: 13 June 2013

https://projecteuclid.org/euclid.aos/1371150894

Digital Object Identifier
doi:10.1214/12-AOS1078

Mathematical Reviews number (MathSciNet)
MR3113804

Zentralblatt MATH identifier
1292.62052

Subjects
Primary: 62F99: None of the above, but in this section

#### Citation

Rinaldo, Alessandro; Petrović, Sonja; Fienberg, Stephen E. Maximum lilkelihood estimation in the $\beta$-model. Ann. Statist. 41 (2013), no. 3, 1085--1110. doi:10.1214/12-AOS1078. https://projecteuclid.org/euclid.aos/1371150894

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#### Supplemental materials

• Supplementary material: Supplement to “Maximum lilkelihood estimation in the $\beta$-model”. In the supplementary material we extend our analysis to other models for network data: the Rasch model, the $\beta$-model with no sampling constraints on the number of observed edges per dyad, the Bradley–Terry model and the $p_{1}$ model of Holland and Leinhardt (1981). We also provide details on how to determine whether a given degree sequence belongs to the interior of the polytope of degree sequences $P_{n}$ and on how to compute the facial set corresponding to a degree sequence on the boundary of $P_{n}$.