提出标准化方法，让加权排名相关系数在随机情况下期望值为零。
Standardization of Weighted Ranking Correlation Coefficients
- 设计通用标准化函数，修复加权相关系数的对称性缺陷。
- 通过蒙特卡洛与多项式回归估算分布参数，适用于长排名场景。
- 适合关注排名重要性差异的研究者，如推荐系统、排序学习。
统计学中衡量两个项目排名相关性的核心问题，传统方法如Kendall's τ和Spearman's ρ具有对称性，保证了在均匀随机排名下的零期望值。但在现代应用中，顶端项目权重更高，促使发展加权变体。这类加权方案破坏原方法的对称性，导致独立排名下期望值非零，影响零相关性的解释。本文提出一个通用标准化函数g(·)，将任意排名相关系数Γ转换为标准化形式g(Γ)，使其在随机情况下期望值为零。该变换保持取值区间[-1,1]，满足边界条件，连续且单调递增，并在原有零期望系数上退化为恒等映射。g(x)的构造依赖于Γ的均值、方差及左方差三个分布参数；由于大排名长度n下精确计算不可行，我们采用蒙特卡洛采样结合多项式回归，获得这些参数随n变化的高精度估计。
原文摘要 · Abstract (English)
A fundamental problem in statistics is measuring the correlation between two rankings of a set of items. Kendall's $τ$ and Spearman's $ρ$ are well established correlation coefficients whose symmetric structure guarantees zero expected value between two rankings randomly chosen with uniform probability. In many modern applications, however, greater importance is assigned to top-ranked items, motivating weighted variants of these coefficients. Such weighting schemes generally break the symmetry of the original formulations, resulting in a non-zero expected value under independence and compromising the interpretation of zero correlation. We propose a general standardization function $g(\cdot)$ that transforms a ranking correlation coefficient $Γ$ into a standardized form $g(Γ)$ with zero expected value under randomness. The transformation preserves the domain $[-1,1]$, satisfies the boundary conditions, is continuous and increasing, and reduces to the identity for coefficients that already satisfy the zero-expected-value property. The construction of $g(x)$ depends on three distributional parameters of $Γ$: its mean, variance, and left variance; since their exact calculation becomes infeasible for large ranking lengths $n$, we develop accurate numerical estimates based on Monte Carlo sampling combined with polynomial regression to capture their dependence on $n$.
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