The median minimizes the sum of absolute differences while the mean minimizes the sum of square distances. What is the function which is minimized by the Hodges-Lehmann location estimator?
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It’s the value of $\Delta$ that minimizes the absolute value of the signed rank statistic W applied to the data after subtracting that $\Delta$. In fact, W becomes 0 after subtracting the HL estimate.
To put the same concept in other words: rank the distances from this center, the sum of the ranks on the left and the sum on the right are as equal as possible.
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1$\begingroup$ Would you please elaborate or send me to the proper reference? I am failing to follow you $\endgroup$ Commented Feb 8, 2021 at 20:52
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$\begingroup$ It's the definition of the HL estimate. I would recommend the book by Hollander and Wolfe. Nonparametric statistical methods. 2nd ed. John Wiley. $\endgroup$– John LCommented Feb 8, 2021 at 21:04
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