A serial version of Hodges and Lehmann’s “6/π result”

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 ≈ 1.910, and that this bound is sharp. We extend this result to the serial case, showing that, when testing against linear (ARMA) serial dependence, the ARE of the Spearman-Wald-Wolfowitz and Kendall rank-based autocorrelations with respect to the van der Waerden or normal-score ones admits a sharp upper bound of (6/pi)2 ≈ 3.648.