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. 
Ting Yan, Yuanzhang Li, Jinfeng Xu, Yaning Yang and Ji Zhu
math.ST stat.TH