Bradley-Terry Model In Sparse Paired Comparisons

The purpose of this dissertation is to study the asymptotic theory of the maximum likelihood estimation in the Bradley-Terry model when the number of parameters goes to infinity and paired comparisons are sparse. In the classical background that the number of parameters is fixed and the number of paired com-parisons in each pair goes to infinity, the asymptotic properties of the maximum likelihood estimation are standard. However, in most applications, the number of subjects is large and the number of paired comparisons in each pair is small and even sparse. For example, consider the National Football League (NFL). It is organized by32teams and these32teams are divided into8divisions. Each division has4teams. In the regular season, each team plays16games (6extra-division games and10inter-division games). In this design, only32×13/2208pairs have comparisons in contrast with the total number of pairs32×31/2496. Therefore asymptotics considering such scenarios are more appealing. Some main contributions of this thesis are as follows:We propose the sparsity conditions for paired comparisons by controlling the length of path from one subject to anoth-er subject in the undirected graph constructed by the design matrix and prove the uniform consistency and asymptotic normality for the maximum likelihood estimation wherein and reveal high dimensions Wilks phenomena for the Bradley-Terry model as well. Extensive simulations and practical data analyses are carried out to demonstrate our theoretical results.