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  "body": "\n## Definition\n\nLet $x_1, \\dots, x_n$ be real numbers.\nThe *Walsh averages* are the $n(n+1)/2$ numbers $(x_i + x_j)/2$ for $1 \\le i \\le j \\le n$.\nThe *one-sample Hodges–Lehmann estimator* is the [[median|08W2CW8PKW]] of the Walsh averages:\n$$\\hat\\theta = \\operatorname{med}\\Big\\{ \\frac{x_i + x_j}{2} : 1 \\le i \\le j \\le n \\Big\\}.$$\nFor two samples $x_1, \\dots, x_m$ and $y_1, \\dots, y_n$, the *two-sample Hodges–Lehmann estimator* of the shift is $\\hat\\Delta = \\operatorname{med}\\set{y_j - x_i : 1 \\le i \\le m, \\ 1 \\le j \\le n}$.\n\n## Remarks\n\nIt is an [[estimator|0TTNRP5PFQ]] of the centre of symmetry $\\theta$ in the model of the [[Wilcoxon signed-rank test|38SETDT2HN]].\nThe 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 $n(n+1)/4$.\nIts asymptotic breakdown point is $1 - 1/\\sqrt{2} \\approx 0.29$.\n\nExample: data $(1, 2, 9)$.\nThe Walsh averages are $1, 2, 9$ (for $i = j$) and $1.5, 5, 5.5$ (for $i \u003c j$); sorted, $1, 1.5, 2, 5, 5.5, 9$.\nTheir median is $(2 + 5)/2 = 3.5$.\n\nNon-example: the median of the averages $(x_i + x_j)/2$ over $i \u003c j$ only, without the terms $i = j$, is a different estimator; for $(1, 2, 9)$ it gives $5$.\nBoth versions appear in the literature, so the convention should be stated.\n",
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