Pseudomedian

In statistics, the pseudomedian is a measure of centrality for data-sets and populations. It agrees with the median for symmetric data-sets or populations. In mathematical statistics, the pseudomedian is also a location parameter for probability distributions.

Description

The pseudomedian of a distribution ${\displaystyle F}$ is defined to be a median of the distribution of ${\displaystyle (Z_{1}+Z_{2})/2}$, where ${\displaystyle Z_{1}}$ and ${\displaystyle Z_{2}}$ are independent, each with the same distribution ${\displaystyle F}$.^[1]

When ${\displaystyle F}$ is a symmetric distribution, the pseudomedian coincides with the median; otherwise this is not generally the case.

The Hodges–Lehmann statistic, defined as the median of all of the midpoints of pairs of observations, is a consistent estimator of the pseudomedian.

Like the set of medians, the pseudomedian is well defined for all probability distributions, even for the many distributions that lack modes or means.

Pseudomedian filter in signal processing

In signal processing there is another definition of pseudomedian filter for discrete signals.

For a time series of length 2N + 1, the pseudomedian is defined as follows. Construct N + 1 sliding windows each of length N + 1. For each window, compute the minimum and maximum. Across all N + 1 windows, find the maximum minimum and the minimum maximum. The pseudomedian is the average of these two quantities.^[2]

See also

• Hodges–Lehmann estimator
• Median filter
• Lulu smoothing

References

1. Hollander, M. and Wolfe, D. A. (2014). Nonparametric Statistical Methods (3nd Ed.). p58
2. W. Pratt, T. Cooper, and I. Kabir. Pseudomedian filter. Architectures and Algorithms for Digital Image Processing II, pages 34–43. Proc. SPIE 534, 1985.