In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the pseudomedian is a measure of
centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key ...
for data-sets and populations. It agrees with the
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
for symmetric data-sets or populations. In
mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
, the pseudomedian is also a
location parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
for
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s.
Description
The pseudomedian of a distribution
is defined to be a median of the distribution of
, where
and
are independent, each with the same distribution
.
When
is a symmetric distribution, the pseudomedian coincides with the
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
; 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
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...
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
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
there is another definition of pseudomedian filter for discrete signals.
For a time series of length 2''N'' + 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.
[W. Pratt, T. Cooper, and I. Kabir. Pseudomedian filter. Architectures and Algorithms for Digital Image Processing II, pages 34–43. Proc. SPIE 534, 1985.]
See also
*
Hodges–Lehmann estimator
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''-distri ...
*
Median filter
The median filter is a non-linear digital filtering technique, often used to remove noise from an image or signal. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an ...
*
Lulu smoothing In signal processing, Lulu smoothing is a nonlinear mathematical technique for removing impulsive Noise (signal processing), noise from a data sequence such as a time series. It is a nonlinear equivalent to taking a moving average (or other smoothin ...
References
Means
Summary statistics
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