  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.
