Unbiased analytic non-parametric correlation estimators in the presence of ties

TL;DR

This paper introduces unbiased, minimum variance non-parametric correlation estimators that effectively handle ties and biases in finite samples, enhancing the reliability of correlation analysis.

Contribution

It develops an inner-product Hilbert space framework to create unbiased estimators for Kendall and Spearman correlations that account for ties and finite sample biases.

Findings

Proposes unbiased estimators with minimal variance for correlation measures.

Shows sub-Gaussian distributions improve estimator performance.

Provides analytically consistent Wald test statistics for finite samples.

Abstract

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.