Wilks' theorems in some exponential random graph models

We are concerned here with the likelihood ratio statistics in two exponential random graph models–the β -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 : β i = β 0 i for i = 1 , . . . , r and H 0 : β 1 = . . . = β r , we show that 2[ (cid:96) ( (cid:98) β ) − (cid:96) ( (cid:98) β 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, (cid:96) ( β ) is the log-likelihood function on the parameter β , (cid:98) β is the MLE under the full parameter space, and (cid:98) β 0 is the restricted MLE under the null parameter space. For two increasing dimensional null hypotheses H 0 : β i = β 0 i for i = 1 , . . . , n and H 0 : β 1 = . . . = β r with r/n ≥ c , we show that the normalized log-likelihood ratio statistics, (2[ (cid:96) ( (cid:98) β ) − (cid:96) ( β 0 )] − n ) / (2 n ) 1 / 2 and (2[ (cid:96) ( (cid:98) β ) − (cid:96) ( (cid:98) β 0 )] − r ) / (2 r ) 1 / 2 , both converge in distribution to the standard normal distribution. Simulation studies and an application to NBA data illustrate the theoretical results.