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 Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of s ...

, 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 Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...

s, as used in the

skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal d ...

and

kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...

, 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 In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

(second) are examples of low-order statistics. HOS are particularly used in the estimation of

shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. ...

s, such as skewness and kurtosis, as when measuring the deviation of a distribution from the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

. 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 In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. Definitions The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional po ...

and

trispectrum In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. Definitions The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditiona ...

. 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 {{statistics-stub