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 term higher-order statistics (HOS) refers to functions which use the third or higher power of a
sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zeroth, first, and second powers). The third and higher
moment
Moment or Moments may refer to:
* Present time
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* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
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s, as used in the
skewness and
kurtosis, are examples of HOS, whereas the first and second moments, as used in the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
(first), and
variance (second) are examples of low-order statistics. HOS are particularly used in the estimation of
shape parameters, such as skewness and kurtosis, as when measuring the deviation of a distribution from the
normal distribution.
In
statistical theory, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of
cumulants and joint cumulants.
[Kendall, MG., Stuart, A. (1969) ''The Advanced Theory of Statistics, Volume 1: Distribution Theory, 3rd Edition'', Griffin. (Chapter 3)] In
time series analysis, the extension of these is to higher order spectra, for example the
bispectrum and
trispectrum.
An alternative to the use of HOS and higher moments is to instead use
L-moments, which are linear statistics (linear combinations of
order statistics), and thus more robust than HOS.
References
External links
*http://www.maths.leeds.ac.uk/Applied/news.dir/issue2/hos_intro.html
*https://web.archive.org/web/20061125033107/http://lpce.cnrs-orleans.fr/~ddwit/lalonde/lalonde_presentations/horbury2.pdf
*http://www.ics.uci.edu/~welling/publications/papers/RobCum-aistats.pdf
Summary statistics
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