Hodges–Lehmann estimator
In
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
Definition
In the simplest case, the "Hodges–Lehmann" statistic estimates the location parameter for a univariate population.[2][3] Its computation can be described quickly. For a dataset with n measurements, the set of all possible 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.
The Hodges–Lehmann statistic also estimates the
Estimating the population median of a symmetric population
For a population that is symmetric, the Hodges–Lehmann statistic estimates the population's median. It is a robust statistic that has a
For symmetric distributions, the Hodges–Lehmann statistic has greater efficiency than does the sample median. For the normal distribution, the Hodges-Lehmann statistic is nearly as efficient as the sample mean. For the Cauchy distribution (Student t-distribution with one degree of freedom), the Hodges-Lehmann is infinitely more efficient than the sample mean, which is not a consistent estimator of the median.[5]
For non-symmetric populations, the Hodges-Lehmann statistic estimates the population's "pseudo-median",
The one-sample Hodges–Lehmann statistic need not estimate any population mean, which for many distributions does not exist. The two-sample Hodges–Lehmann estimator need not estimate the difference of two means or the difference of two (pseudo-)medians; rather, it estimates the differences between the population of the paired random–variables drawn respectively from the populations.[4]
In general statistics
The Hodges–Lehmann univariate statistics have several generalizations in multivariate statistics:[10]
- Multivariate ranks and signs[11]
- Spatial sign tests and spatial medians[8]
- Spatial signed-rank tests[12]
- Comparisons of tests and estimates[13]
- Several-sample location problems[14]
See also
Notes
- ^ Lehmann (2006, pp. 176 and 200–201)
- ISBN 0-19-850994-4Entry for "Hodges-Lehmann one-sample estimator"
- ^ Hodges & Lehmann (1963)
- ^ a b Everitt (2002) Entry for "Hodges-Lehmann estimator"
- ^ a b Myles Hollander. Douglas A. Wolfe. Nonparametric statistical methods. 2nd ed. John Wiley.
- ^ Jureckova Sen. Robust Statistical Procedures.
- ^ Hettmansperger & McKean (1998, pp. 2–4)
- ^ a b Oja (2010, p. 71)
- ^ Hettmansperger & McKean (1998, pp. 2–4 and 355–356)
- ^ Oja (2010, pp. 2–3)
- ^ Oja (2010, p. 34)
- ^ Oja (2010, pp. 83–94)
- ^ Oja (2010, pp. 98–102)
- ^ Oja (2010, pp. 160, 162, and 167–169)
References
- Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-X
- Hettmansperger, T. P.; McKean, J. W. (1998). Robust nonparametric statistical methods. Kendall's Library of Statistics. Vol. 5 (First ed., rather than Taylor and Francis (2010) second ed.). London; New York: Edward Arnold; John Wiley and Sons, Inc. pp. xiv+467. MR 1604954.
- Hodges, J. L.; .
- MR 0395032.
- Oja, Hannu (2010). Multivariate nonparametric methods with R: An approach based on spatial signs and ranks. Lecture Notes in Statistics. Vol. 199. New York: Springer. pp. xiv+232. MR 2598854.
- Zbl 0119.15604.