Wilks' theorems in the $\beta$-model

Likelihood ratio tests and the Wilks theorems have been pivotal in statistics but have rarely been explored in network models with an increasing dimension. We are concerned here with likelihood ratio tests in the $\beta$-model for undirected graphs. For two growing dimensional null hypotheses including a specified null $H_0: \beta_i=\beta_i^0$ for $i=1,\ldots, r$ and a homogenous null $H_0: \beta_1=\cdots=\beta_r$, we reveal high dimensional Wilks' phenomena that the normalized log-likelihood ratio statistic, $[2\{\ell(\widehat{\boldsymbol{\beta}}) - \ell(\widehat{\boldsymbol{\beta}}^0)\} - r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution as $r$ goes to infinity. Here, $\ell( \boldsymbol{\beta})$ is the log-likelihood function on the vector parameter $\boldsymbol{\beta}=(\beta_1, \ldots, \beta_n)^\top$, $\widehat{\boldsymbol{\beta}}$ is its maximum likelihood estimator (MLE) under the full parameter space, and $\widehat{\boldsymbol{\beta}}^0$ is the restricted MLE under the null parameter space. For the corresponding fixed dimensional null $H_0: \beta_i=\beta_i^0$ for $i=1,\ldots, r$ and the homogenous null $H_0: \beta_1=\cdots=\beta_r$ with a fixed $r$, we establish Wilks type of results that $2\{\ell(\widehat{\boldsymbol{\beta}}) - \ell(\widehat{\boldsymbol{\beta}}^0)\}$ converges in distribution to a Chi-square distribution with respective $r$ and $r-1$ degrees of freedom, as the total number of parameters, $n$, goes to infinity. The Wilks type of results are further extended into a closely related Bradley--Terry model for paired comparisons, where we discover a different phenomenon that the log-likelihood ratio statistic under the fixed dimensional specified null asymptotically follows neither a Chi-square nor a rescaled Chi-square distribution. Simulation studies and an application to NBA data illustrate the theoretical results.