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Abstract One version of the Hodges-Lehmann estimator, = median (Xi + Xj) /2, 1 ≤ i, j ≤ n, is also the minimum distance estimator derived from a particular Cramér-von Mises distance. This distance evaluated at, that is, the minimized distance, provides a natural statistic for testing asymmetry of the underlying distribution.
Dennis D. Boos (1982) studied this question.