Higher-order statistics
Encyclopedia
Higher-order statistics (HOS) are descriptive measures of, among other things, qualities of probability distribution
s and sample distributions, and are, themselves, extensions of first- and second-order measures (such as the mean
, variance
, autocorrelation function, and power spectrum) to higher orders. Skewness
and kurtosis
are examples of this. Arguably, the estimates of coefficients and associated significance tests, in a regression analysis
, of terms for effects higher than quadratic
could be included here, too.
In statistical theory
, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of cumulant
s and joint cumulants
. In time series analysis, the extension of these is to higher order spectra, for example the bispectrum
and trispectrum
.
An alternative to generalising the use of moments is to consider L-moment
s which allow higher-order statistics to be based on linear combinations of order statistic
s.
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s and sample distributions, and are, themselves, extensions of first- and second-order measures (such as the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
, variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
, autocorrelation function, and power spectrum) to higher orders. Skewness
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...
and kurtosis
Kurtosis
In probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable...
are examples of this. Arguably, the estimates of coefficients and associated significance tests, in a regression analysis
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
, of terms for effects higher than quadratic
Quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
could be included here, too.
In statistical theory
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 statistical inference, and the actions and deductions that...
, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of cumulant
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
s and joint cumulants
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
. In time series analysis, the extension of these is to higher order spectra, for example the bispectrum
Bispectrum
In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum...
and trispectrum
Trispectrum
In mathematics, in the area of statistical analysis, the trispectrum is a statistic used to search for nonlinear interactions. The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum...
.
An alternative to generalising the use of moments is to consider L-moment
L-moment
In statistics, L-moments are statistics used to summarize the shape of a probability distribution. They are analogous to conventional moments in that they can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis...
s which allow higher-order statistics to be based on linear combinations of order statistic
Order statistic
In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference....
s.
External links
- http://www.maths.leeds.ac.uk/Applied/news.dir/issue2/hos_intro.html
- http://lpce.cnrs-orleans.fr/~ddwit/lalonde/lalonde_presentations/horbury2.pdf
- http://www.ics.uci.edu/~welling/publications/papers/RobCum-aistats.pdf