A serial version of Hodges and Lehmann’s “6/π result”
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Date
2013-04-08
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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.
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Keywords
asymptotic relative efficiency, Kendall autocorrelations, linear serial rank statistics, rank-based tests, Spearman autocorrelations, van der Waerden test, Wilcoxon test