Asymptotic representations for Spearman's footrule correlation coefficient

TL;DR

This paper develops two asymptotic representations for Spearman's footrule correlation coefficient to better understand its distribution under independence, supported by simulations showing their accuracy.

Contribution

It introduces novel asymptotic representations that simplify and rigorously justify the distributional properties of Spearman's footrule coefficient under independence.

Findings

Two asymptotic representations accurately approximate the distribution.

Simulation results confirm the effectiveness of the proposed methods.

The methods justify asymptotic normality of the coefficient.

Abstract

In order to address the theoretical challenges arising from the dependence structure of ranks in Spearman's footrule correlation coefficient, we propose two asymptotic representations to approximate the distribution of this coefficient under the hypothesis of independence. The first representation simplifies the dependence structure by replacing empirical distribution functions with their population counterparts. The second representation leverages the H\'{a}jek projection technique to decompose the initial form into a sum of independent components, thereby rigorously justifying asymptotic normality. Simulation studies demonstrate the appropriateness of two proposed asymptotic representations, as well as their excellent approximation to the limiting normal distribution.