Asymptotic theory of sparse Bradley–Terry model

The Bradley–Terry model is a fundamental model in the analysis of network data involving paired comparison. Assuming every pair of subjects in the network have an equal number of comparisons, Simons and Yao (Ann. Statist. 27 (1999) 1041–1060) established an asymptotic theory for statistical estimation in the Bradley–Terry model. In practice, when the size of the network becomes large, the paired comparisons are generally sparse. The sparsity can be characterized by the probability $p_{n}$ that a pair of subjects have at least one comparison, which tends to zero as the size of the network $n$ goes to infinity. In this paper, the asymptotic properties of the maximum likelihood estimate of the Bradley–Terry model are shown under minimal conditions of the sparsity. Specifically, the uniform consistency is proved when $p_{n}$ is as small as the order of $(\log n)^{3}/n$, which is near the theoretical lower bound $\log n/n$ by the theory of the Erdos–Renyi graph. Asymptotic normality and inference are also provided. Evidence in support of the theory is presented in simulation results, along with an application to the analysis of the ATP data.