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

We are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a “merit vector” (u₀,,uₜ), representing the merits of $t +1$ teams (players, treatments, objects), under the Bradley–Terry model, as t → ∞. 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 uᵢ and uⱼ play each other, the probability that team i prevails is assumed to be uᵢ/(uᵢ + uⱼ). Suppose each pair of teams play each other exactly n times for some fixed n. The objective is to estimate the merits, uᵢ’s, based on the outcomes of the $nt(t +1)/2$ games. Clearly, the model depends on the uᵢ’s only through their ratios. Under some condition on the growth rate of the largest ratio uᵢ/uⱼ (0 ≤ i, j ≤ t) as t → ∞, the maximum likelihood estimator of (u₁/u₀,,uₜ/u₀) is shown to be consistent and asymptotically normal. Some simulation results are provided.