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 (u 0 , ..., u t ), 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 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 < i, j < t) as t → ∞, the maximum likelihood estimator of (u 1 /u 0 u t /u 0 ) is shown to be consistent and asymptotically normal. Some simulation results are provided.