On Hodges and Lehmann's "$6/\pi$ result"

TL;DR

This paper revisits Hodges and Lehmann's 1961 result on the upper bound of the asymptotic relative efficiency of Wilcoxon tests compared to normal-score tests, extending the analysis to Student score-based statistics and autocorrelation measures.

Contribution

The paper extends Hodges and Lehmann's bounded efficiency result to Student score-based statistics and autocorrelation tests, providing new bounds under various densities.

Findings

Bounded ARE of Wilcoxon tests relative to normal-score tests is approximately 1.91.

Derived efficiency bounds for Student score-based autocorrelation statistics.

Analyzed the serial version of the ARE for various densities.

Abstract

While the asymptotic relative efficiency (ARE) of Wilcoxon rank-based tests for location and regression with respect to their parametric Student competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that the ARE of the same Wilcoxon tests with respect to their van der Waerden or normal-score counterparts is bounded from above by $6/\pi\approx 1.910$. In this paper, we revisit that result, and investigate similar bounds for statistics based on Student scores. We also consider the serial version of this ARE. More precisely, we study the ARE, under various densities, of the Spearman-Wald-Wolfowitz and Kendall rank-based autocorrelations with respect to the van der Waerden or normal-score ones used to test (ARMA) serial dependence alternatives.