Wilks' theorems in some exponential random graph models

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.