MAXIMUM LILKELIHOOD ESTIMATION IN THE β-MODEL

• 成果类型: Article
• 署名作者: Rinaldo, Alessandro; Petrovic, Sonja; Fienberg, Stephen E.
• 署名单位: Carnegie Mellon University; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park; Carnegie Mellon University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1078
• 发表日期: 2013
• 页码: 1085-1110
• 关键词: probability-distributions; exponential-families; expotential family; directed-graphs

摘要

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.