Hodges-Lehmann estimator

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The Hodges-Lehmann estimator is a statistical method for robust esimation.

The principal form of this estimator is used to give an estimate of the difference between the values in two sets of data. If the two sets of data contain m and n data points respectively, $m \times n$ pairs of points (one from each set) can be formed and each pair gives a difference of values. The Hodges-Lehmann estimator for the difference is defined as the median of the $m \times n$ differences.

A second type of estimate which has also been called by the name "Hodges-Lehmann" relates to defining a location estimate for a single dataset. In this case, if the dataset contains n data points, it is possible to define $n (n + 1) / 2$ pairs within the data set, allowing each item to pair with itself. The average value is calculated for each pair and the final estimate of location is the median of the $n (n + 1) / 2$ averages.

Note that the two-sample Hodges-Lehmann estimator does not estimate the difference of the means or the difference of the medians (it estimates the median of the differences, which, if the underlying distributions are asymmetric, is a different quantity), while the one-sample Hodges-Lehmann estimator does not estimate either the mean or the median.