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June 2013 Maximum lilkelihood estimation in the $\beta$-model
Alessandro Rinaldo, Sonja Petrović, Stephen E. Fienberg
Ann. Statist. 41(3): 1085-1110 (June 2013). DOI: 10.1214/12-AOS1078


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.


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Alessandro Rinaldo. Sonja Petrović. Stephen E. Fienberg. "Maximum lilkelihood estimation in the $\beta$-model." Ann. Statist. 41 (3) 1085 - 1110, June 2013.


Published: June 2013
First available in Project Euclid: 13 June 2013

zbMATH: 1292.62052
MathSciNet: MR3113804
Digital Object Identifier: 10.1214/12-AOS1078

Primary: 62F99

Keywords: $\beta$-model , maximum likelihood estimator , polytope of degree sequences , Random graphs

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 3 • June 2013
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