In statistics, 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 moments, 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 ...

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, kurt ...

, are examples of HOS, whereas the first and second moments, as used in the arithmetic mean (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 numbe ...

(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. t ...

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 The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistica ...

, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of

cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...

s 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

, 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 p ...

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-moment In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics ( L-statistics) analogous to conventional moments, and can be used to calculate qu ...

s, which are linear statistics (linear combinations of

order statistic In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Importan ...

s), 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