An unbiased non-parametric correlation estimator in the presence of ties

An inner-product Hilbert space formulation of the Kemeny distance is defined over the domain of all permutations with ties upon the extended real line, and results in an unbiased minimum variance (Gauss-Markov) correlation estimator upon a homogeneous i.i.d. sample. In this work, we construct and prove the necessary requirements to extend this linear topology for both Spearman's $\rho$ and Kendall's $\tau_{b}$, showing both spaces to be both biased and inefficient upon practical data domains. A probability distribution is defined for the Kemeny $\tau_{\kappa}$ estimator, and a Studentisation adjustment for finite samples is provided as well. This work allows for a general purpose linear model duality to be identified as a unique consistent solution to many biased and unbiased estimation scenarios.