Hodges–Lehmann estimate

In statistics, the Hodges–Lehmann estimator is a robust and nonparametric estimator of a population's location parameter. For populations that are symmetric about one median, such as the (Gaussian) normal distribution or the Student t-distribution, the Hodges–Lehmann estimator is a consistent and median-unbiased estimate of the population median. For non-symmetric populations, the Hodges–Lehmann estimator estimates the "pseudo–median", which is closely related to the population median.

The Hodges–Lehmann estimator was proposed originally for estimating the location parameter of one-dimensional populations, but it has been used for many more purposes. It has been used to estimate the differences between the members of two populations. It has been generalized from univariate populations to multivariate populations, which produce samples of vectors.

It is based on the Wilcoxon signed-rank statistic. In statistical theory, it was an early example of an rank-based estimator, an important class of estimators both in nonparametric statistics and in robust statistics. The Hodges–Lehmann estimator was proposed in 1963 independently by Pranab Kumar Sen and by Joseph Hodges and Erich Lehmann, and so it is also called the "Hodges–Lehmann–Sen estimator".

In the simplest case, the "Hodges–Lehmann" statistic estimates the location parameter for a univariate population. Its computation can be described quickly. For a dataset with n measurements, the set of all possible one- or two-element subsets of it has n(n + 1)/2 elements. For each such subset, the mean is computed; finally, the median of these n(n + 1)/2 averages is defined to be the Hodges–Lehmann estimator of location.

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