An unbiased minimum variance non-parametric analytic and likelihood estimator for discrete and continuous score spaces

This manuscript develops a general purpose inner-product norm for the Kendall $\tau$ and Spearman's $\rho$, which operates as an unbiased MLE even in the presence of ties. We derive and prove the strict sub-Gaussianity of the Kemeny norm-space, thereby disproving conclusions developed by both \textcite{kendall1948} and \textcite{diaconis1977} as to the nature of the appropriate, finite sample, probability distribution and test statistics. A non-parametric MLE framework for all bivariate pairs is developed, thereby resolving an hypothesis of \textcite{olkin1994} concerning an exponential multivariate distribution for order statistics, by showing that for finite samples, the distribution is non-exponential. Non-parametric linear estimators are also constructed for the polychoric correlations and by extension, a linearly decomposable non-parametric multidimensional linear system of equations for non-parametric Factor Analysis is shown.