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

• 成果类型: Article
• 署名作者: Simons, G; Yao, YC
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Academia Sinica - Taiwan
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1999
• 页码: 1041-1060
• 关键词: p-regression parameters; mathematicians guide; popular sports; M-ESTIMATORS; BEHAVIOR; p2/n

摘要

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 --> infinity. 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 less than or equal to i, j less than or equal to t) as t --> infinity, 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.