Monotonicity properties of the asymptotic relative efficiency between common correlation statistics in the bivariate normal model

TL;DR

This paper investigates how the asymptotic relative efficiency between Pearson's, Spearman's, and Kendall's correlation statistics varies monotonically with the population correlation in bivariate normal samples, revealing a fundamental property.

Contribution

It establishes the monotonic relationship of asymptotic relative efficiency among common correlation measures in the bivariate normal model, using novel monotonicity proof techniques.

Findings

Efficiency ratios depend monotonically on the population correlation

The result applies specifically to bivariate normal i.i.d. samples

Proofs utilize l'Hospital-type rules for monotonicity

Abstract

Pearson's is the most common correlation statistic, used mainly in parametric settings. Most common among nonparametric correlation statistics are Spearman's and Kendall's. We show that for bivariate normal i.i.d. samples the pairwise asymptotic relative efficiency between these three statistics depends monotonically on the population correlation coefficient. This monotonicity is a corollary to a stronger result. The proofs rely on the use of l'Hospital-type rules for monotonicity patterns.