Simultaneous computation of Kendall’s tau and its jackknife variance

Abstract

We present efficient algorithms for simultaneously computing Kendall’s tau and the jackknife estimator of its variance. For the classical pairwise tau, we describe a modification of Knight’s algorithm (originally designed to compute only tau) that does so while preserving its $O\left(n{log}_{2}n\right)$ runtime in the number of observations $n$. We also introduce a novel algorithm computing a multivariate extension of tau and its jackknife variance in $O\left(n{log}_2^{p}n\right)$ time.