We consider the problem of ranking $n$ players from partial pairwise comparison data under the Bradley-Terry-Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendall's tau distance that measures the difference between two rank vectors by counting the number of inversions. The minimax rate of ranking exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. To achieve the minimax rate, we propose a divide-and-conquer ranking algorithm that first divides the $n$ players into groups of similar skills and then computes local MLE within each group. The optimality of the proposed algorithm is established by a careful approximate independence argument between the two steps.

翻译：我们考虑的是,在Bradley-Terry-Luce模式下,从部分对称比较数据中将玩家排位为一美元的问题。在文献中,这一排名问题的最低比率第一次是从Kendall的Tau距离中得出的,该距离通过计算倒数来测量两个阶梯矢量之间的差异。最低的排名率显示指数率和多音率之间的过渡,这取决于问题的信号对噪音比率。据我们所知,这种现象是完全排名所独有的,在任何其他统计估计问题中都看不出来。为了达到迷你轴率,我们建议了一种分而错的排序算法,首先将牌玩家分为类似技能的一组,然后在每个组中进行本地 MLE。提议的算法的最佳性是通过两个步骤之间谨慎的近似独立争论来确定的。