Pseudomedian

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.

In signal processing there is another definition of pseudomedian filter for discrete signal. For a window of width 2N + 1 pseudomedian defined as the average of the maximum of the minima and the minimum of the maxima of the N + 1 sliding subwindows of length N + 1.[2]

• Hodges–Lehmann estimator
• Median filter
• Lulu smoothing