HodgesLehmann

The Hodges–Lehmann estimator is a robust estimator of a population location parameter. It is defined as the median of the set of all pairwise averages of the sample observations: (X_i + X_j)/2 for 1 ≤ i ≤ j ≤ n. This construction makes the estimator less sensitive to outliers than the sample mean while preserving good performance under a wide range of distributions.

Computation involves generating all pairwise averages from the sample and selecting their median. For a sample of size n, this yields n(n+1)/2 values (if i ≤ j is used) and requires sorting to obtain the median. In practice, efficient algorithms exist, but the calculation is more intensive than the mean, especially for large datasets.

Properties of the Hodges–Lehmann estimator include robustness to outliers and, under normal distributions, relatively high efficiency compared with the mean. It is affine equivariant and consistent for the population location parameter. The estimator often lies between the sample median and the mean, providing a compromise between resistance to extreme values and statistical efficiency.

Applications and relationships: the HL estimator is widely used in nonparametric statistics as a robust alternative to the mean for estimating central tendency. In two-sample problems, a related Hodges–Lehmann statistic considers the median of all pairwise differences between samples, serving as a robust measure of location shift.

History: the estimator was introduced by Hodges and Lehmann in the mid-20th century as part of early work on robust estimation of location. It remains a standard reference in nonparametric and robust statistical methods.