Wilks' theorems in some exponential random graph models

TL;DR

This paper establishes Wilks' theorems for likelihood ratio tests in certain exponential random graph models, demonstrating their asymptotic chi-square and normal distributions under fixed and increasing dimensions, supported by simulations and NBA data analysis.

Contribution

It proves Wilks' theorems for the $eta$-model and Bradley-Terry model in high-dimensional settings, extending classical results to growing parameter spaces.

Findings

Likelihood ratio statistics converge to chi-square distributions in fixed dimensions.

Normalized likelihood ratios tend to normal distributions in high dimensions.

Simulation and NBA data confirm theoretical asymptotic behaviors.

Abstract

We are concerned here with the likelihood ratio statistics in two exponential random graph models -- the $\beta$-model and the Bradley-Terry model, in which the degree sequence on an undirected graph and the out-degree sequence on a weighted directed graph are the exclusively sufficient statistics in the exponential-family distributions on graphs, respectively. We prove the Wilks type of theorems for some fixed and growing dimensional hypothesis testing problems. More specifically, under two fixed dimensional null hypotheses $H_0: \beta_i=\beta_i^0$ for $i=1,\ldots, r$ and $H_0: \beta_1=\ldots=\beta_r$, we show that $2[\ell(\widehat{\boldsymbol{\beta}}) - \ell(\widehat{\boldsymbol{\beta}}^0)]$ converges in distribution to a Chi-square distribution with the respective degrees of freedoms, $r$ and $r-1$, as the dimension $n$ of the full parameter space goes to infinity. Here, $\ell(\boldsymbol{\beta})$ is the log-likelihood function on the parameter $\boldsymbol{\beta}$, $\widehat{\boldsymbol{\beta}}$ is the MLE under the full parameter space, and $\widehat{\boldsymbol{\beta}}^0$ is the restricted MLE under the null parameter space. For two increasing dimensional null hypotheses $H_0: \beta_i = \beta_i^0$ for $i=1, \ldots, n$ and $H_0: \beta_1=\ldots=\beta_r$ with $r/n \ge c$, we show that the normalized log-likelihood ratio statistics, $(2[\ell(\widehat{\boldsymbol{\beta}}) - \ell(\boldsymbol{\beta}^0)] -n)/(2n)^{1/2}$ and $(2[\ell(\widehat{\boldsymbol{\beta}}) - \ell(\widehat{\boldsymbol{\beta}}^0)] -r)/(2r)^{1/2}$, both converge in distribution to the standard normal distribution. Simulation studies and an application to NBA data illustrate the theoretical results.