Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons

Abstract

We are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a “merit vector” $(u_0,\dots,u_t)$, representing the merits of $t +1$ teams (players, treatments, objects), under the Bradley–Terry model, as $t \to \infty$. This situation contrasts with the well-known Neyman–Scott problem under which the number of parameters grows with $t$ (the amount of sampling), and for which the maximum likelihood estimator fails even to attain consistency. A key feature of our proof is the use of an effective approximation to the inverse of the Fisher information matrix. Specifically, under the Bradley–Terry model, when teams $i$ and $j$ with respective merits $u_i$ and $u_j$ play each other, the probability that team $i$ prevails is assumed to be $u_i/(u_i + u_j)$. Suppose each pair of teams play each other exactly $n$ times for some fixed $n$. The objective is to estimate the merits, $u_i$’s, based on the outcomes of the $nt(t +1)/2$ games. Clearly, the model depends on the $u_i$’s only through their ratios. Under some condition on the growth rate of the largest ratio $u_i/u_j (0 \leq i, j \leq t)$ as $t \to \infty$, the maximum likelihood estimator of $(u_1/u_0,\dots,u_t/u_0)$ is shown to be consistent and asymptotically normal. Some simulation results are provided.