An inner-product Hilbert space formulation is defined over a domain of all permutations with ties upon the extended real line. We demonstrate this work to resolve the common first and second order biases found in the pervasive Kendall and Spearman non-parametric correlation estimators, while presenting as unbiased minimum variance (Gauss-Markov) estimators. We conclude by showing upon finite samples that a strictly sub-Gaussian probability distribution is to be preferred for the Kemeny $\tau_{\kappa}$ and $\rho_{\kappa}$ estimators, allowing for the construction of expected Wald test statistics which are analytically consistent with the Gauss-Markov properties upon finite samples.

翻译：[翻译摘要] 本文在扩展实直线上的全排列（含结）域上定义了内积希尔伯特空间框架。我们证明该方法能消除普遍使用的Kendall秩相关系数与Spearman秩相关系数中常见的一阶和二阶偏差，同时给出无偏最小方差（高斯-马尔可夫）估计量。最后通过有限样本分析表明，Kemeny $\tau_{\kappa}$ 与 $\rho_{\kappa}$ 估计量应优先采用严格次高斯概率分布，从而构建出有限样本下与高斯-马尔可夫性质解析一致的期望Wald检验统计量。