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Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters

Maximum entropy models, motivated by applications in neuron science, are natural generalizations of the $β$-model to weighted graphs. Similar to the $β$-model, each vertex in maximum entropy models is assigned a potential parameter, and the degree sequence is the natural sufficient statistic. Hillar and Wibisono (2013) has proved the consistency of the maximum likelihood estimators. In this paper, we further establish the asymptotic normality for any finite number of the maximum likelihood estimators in the maximum entropy models with three types of edge weights, when the total number of parameters goes to infinity. Simulation studies are provided to illustrate the asymptotic results.