Hodges–Lehmann Estimator | Andrey Akinshin

The Hodges–Lehmann Estimator is a robust measure of location and location shift. Introduced in hodges1963 and sen1963.

For a single sample $\mathbf{x} = ( x_1, x_2, \ldots, x_n )$, the Hodges-Lehmann location estimator (known as pseudo-median) is defined as the median of the Walsh (pairwise) averages:

$$ \operatorname{HL}(\mathbf{x}) = \underset{1 \leq i \leq j \leq n}{\operatorname{median}} \left(\frac{x_i + y_j}{2} \right). $$

For two samples $\mathbf{x} = ( x_1, x_2, \ldots, x_n )$ and $\mathbf{y} = ( y_1, y_2, \ldots, y_m )$, the Hodges-Lehmann location shift estimator is defined as follows:

$$ \operatorname{HL}(\mathbf{x}, \mathbf{y}) = \underset{1 \leq i \leq n,\,\, 1 \leq j \leq m}{\operatorname{median}} \left(x_i - y_j \right). $$

Asymptotic breakdown point: $\approx 29\%$; asymptotic Gaussian efficiency: $\approx 96\%$.

Reference R implementation (the default implementation is buggy):

hl <- function(x, y = NULL) {
  if (is.null(y)) {
    walsh <- outer(x, x, "+") / 2
    median(walsh[lower.tri(walsh, diag = TRUE)])
  } else {
    median(outer(x, y, "-"))
  }
}

Posts (18) Papers (3)

Posts (18)

Hodges-Lehmann ratio estimator vs. Bhattacharyya's scale ratio estimator (2023-12-26) 2
Resistance to the low-density regions: the Hodges-Lehmann location estimator based on the Harrell-Davis quantile estimator (2023-11-21) 2
Median vs. Hodges-Lehmann: compare efficiency under heavy-tailedness (2023-11-14) 2 1
Weighted Hodges-Lehmann location estimator and mixture distributions (2023-10-03) 2
Hodges-Lehmann Gaussian efficiency: location shift vs. shift of locations (2023-09-12) 2
Ratio estimator based on the Hodges-Lehmann approach (2023-08-29) 1
Understanding the pitfalls of preferring the median over the mean (2023-06-20)
Efficiency of the central tendency measures under the uniform distribution (2023-05-16)
Unobvious problems of using the R's implementation of the Hodges-Lehmann estimator (2023-05-09)
Weighted modification of the Hodges-Lehmann location estimator (2023-03-28) 1 3
Trimmed Hodges-Lehmann location estimator, Part 2: Gaussian efficiency (2023-01-10) 1
Trimmed Hodges-Lehmann location estimator, Part 1: breakdown point (2023-01-03) 1
Median of the shifts vs. shift of the medians, Part 2: Gaussian efficiency (2022-12-27) 1 1
Median of the shifts vs. shift of the medians, Part 1 (2022-12-20) 2
Resistance to the low-density regions: the Hodges-Lehmann location estimator (2022-12-13) 2 1
Hodges-Lehmann-Sen shift and shift confidence interval estimators (2022-05-31) 5
Statistical efficiency of the Hodges-Lehmann median estimator, Part 2 (2022-05-24) 3 3
Statistical efficiency of the Hodges-Lehmann median estimator, Part 1 (2022-05-17) 3 2

Library / Papers (3)

A Note on the Hodges–Lehmann Estimator (2010) by Gerd K Rosenkranz
Estimates of Location Based on Rank Tests (1963) by J L Hodges et al. 1
Nonparametric Confidence Intervals for a Shift Parameter (2011) by E. L. Lehmann

Backlinks (2)

Effect Sizes and Asymmetry (2024-03-12) 5
Statistical efficiency of the Hodges-Lehmann median estimator, Part 1 (2022-05-17) 3 2