Hodges–Lehmann estimator

Remarks

It is an estimator of the centre of symmetry $\theta$ in the model of the Wilcoxon signed-rank test. The estimate $\hat\theta$ is the value of $t$ for which the signed-rank statistic of $x_1 - t, \dots, x_n - t$ is closest to its null mean $n(n+1)/4$. Its asymptotic breakdown point is $1 - 1/\sqrt{2} \approx 0.29$.

Example: data $(1, 2, 9)$. The Walsh averages are $1, 2, 9$ (for $i = j$) and $1.5, 5, 5.5$ (for $i < j$); sorted, $1, 1.5, 2, 5, 5.5, 9$. Their median is $(2 + 5)/2 = 3.5$.

Non-example: the median of the averages $(x_i + x_j)/2$ over $i < j$ only, without the terms $i = j$, is a different estimator; for $(1, 2, 9)$ it gives $5$. Both versions appear in the literature, so the convention should be stated.