The Hodges-Lehmann estimator provides, in the one-sample case, an estimate of the center of a distribution, and in the two-sample case, an estimate of the difference in the centers of two distributions.
The one-sample estimator is defined as the median of the set of \(n\left (n + 1\right )/2\) Walsh averages. Each Walsh average is the arithmetic average of two observations. For example, consider the set of 5 observations (1, 3, 8, 9, 15). Table 1 shows the computation of the \(5\left (5 + 1/2\right ) = 15\) Walsh averages.
The median of the Walsh averages is the one-sample Hodges-Lehmann estimator of the center of the distribution. In this example, the median of the 15 Walsh averages (the Hodges-Lehmann estimator) is 8. Note that in this case, the Hodges-Lehmann estimator is equal to the simple median of the original five observations, which is also 8. Of course, the Hodges-Lehmann estimator does not have to necessarily equal the sample median. While both the median and Hodges-Lehmann...
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References and Further Reading
Lehmann EL (1999) Elements of large sample theory. Springer, New York
Lehmann EL (1998) Nonparametrics: statistical methods based on ranks. Prentice Hall, Upper Saddle River
Sprent P (1998) Data driven statistical methods. Chapman & Hall, London
Walsh JE (1949) Some significance tests for the median which are valid under very general conditions. Ann Math Stat 20:64–81
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Hershberger, S.L. (2011). Hodges-Lehmann Estimators. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_290
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