Simultaneous computation of Kendall's tau and its jackknife variance

TL;DR

This paper introduces efficient algorithms for computing Kendall's tau and its jackknife variance simultaneously, including a multivariate extension, maintaining optimal computational complexity.

Contribution

It presents a modified Knight's algorithm for classical tau and a new algorithm for multivariate tau with jackknife variance, both optimized for speed.

Findings

Classical tau computation achieved in O(n log n) time.

Multivariate tau and variance computed in O(n log^p n) time.

Algorithms preserve efficiency while extending to multivariate cases.

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 $O(n \log_2 n)$ runtime in the number of observations $n$. We also introduce a novel algorithm computing a multivariate extension of tau and its jackknife variance in $O(n \log_2^p n)$ time.