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

TL;DR

This paper proves that maximum likelihood estimators in maximum entropy models for weighted graphs become normally distributed as the number of parameters increases, extending previous consistency results.

Contribution

It establishes the asymptotic normality of MLEs in maximum entropy models with increasing parameters, for various edge weight types.

Findings

MLEs are asymptotically normal as parameters grow

Simulation studies confirm theoretical results

Results extend understanding of maximum entropy models in graph analysis

Abstract

Maximum entropy models, motivated by applications in neuron science, are natural generalizations of the $\beta$-model to weighted graphs. Similar to the $\beta$-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.