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Encyclopedia > Wishart distribution
Wishart
Probability density function
Cumulative distribution function
Parameters n > 0! deg. of freedom (real)
mathbf{V} > 0, scale matrix ( pos. def)
Support mathbf{V}! is positive definite
Probability density function (pdf) frac{left|mathbf{W}right|^frac{n-p-1}{2}} {2^frac{np}{2}left|{mathbf V}right|^frac{n}{2}Gamma_p(frac{n}{2})} expleft(-frac{1}{2}{rm Tr}({mathbf V}^{-1}mathbf{W})right)
Cumulative distribution function (cdf)
Mean nΣ
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function Theta mapsto left|{mathbf I} - 2i,{mathbfTheta}{mathbfSigma}right|^{-n/2}

In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables ("random matrices"). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. Please refer to Real vs. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... John Wishart (28 November 1898 – 14 July 1956) was a Scottish agricultural statistician. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... For the square matrix section, see square matrix. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. ... Multivariate statistics or multivariate statistical analysis in statistics describes a collection of procedures which involve observation and analysis of more than one statistical variable at a time. ...

Contents

Definition

Suppose X is an ntimes p matrix each row of which is drawn from p-variate normal distribution with zero mean: In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ...

X_{(i)}{=}(x_i^1,...,x_i^p)^Tsim N_p(0,V),

Further suppose that the rows X(1), ..., X(n) are independent. Then the Wishart distribution is the probability distribution of the p×p random matrix In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...

S = XTX,

where S is known as the scatter matrix. One indicates that S has that probability distribution by writing It has been suggested that this article or section be merged into sample mean and sample covariance. ...

Ssim W_p(V,n).

The positive integer n is the number of degrees of freedom. Sometimes this is written W(V,p,n). This article or section is in need of attention from an expert on the subject. ...


If p = 1 and V = 1 then this distribution is a chi-square distribution with n degrees of freedom. In probability theory and statistics, the chi-square distribution (also chi-squared or χ2  distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ...


Occurrence

The Wishart distribution arises frequently in likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of random matrices. A likelihood-ratio test is a statistical test relying on a test statistic computed by taking the ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum with that constraint relaxed. ... In probability theory and statistics, a random matrix is a matrix-valued random variable. ...


Probability density function

The Wishart distribution can be characterized by its probability density function, as follows. In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...


Let {mathbf W} be a ptimes p symmetric matrix of random variables that is positive definite. Let {mathbf V} be a (fixed) positive definite matrix of size ptimes p. In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...


Then, if ngeq p, {mathbf W} has a Wishart distribution with n degrees of freedom if it has a probability density function f_{mathbf W} given by In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

f_{mathbf W}(w)= frac{ left|wright|^{(n-p-1)/2} expleft[ - {rm trace}({mathbf V}^{-1}w/2 )right] }{ 2^{np/2}left|{mathbf V}right|^{n/2}Gamma_p(n/2) }

where Gamma_p(cdot) is the multivariate gamma function defined as In mathematics, the multivariate gamma distribution , , is a generalization of the gamma function. ...

Gamma_p(n/2)= pi^{p(p-1)/4}Pi_{j=1}^p Gammaleft[ (n+1-j)/2right].

In fact the above definition can be extended to any real n > p − 1.


Characteristic function

The characteristic function of the Wishart distribution is Some mathematicians use the phrase characteristic function synonymously with indicator function. ...

Theta mapsto left|{mathbf I} - 2i,{mathbfTheta}{mathbfSigma}right|^{-n/2}.

In other words,

Theta mapsto {mathcal E}left{mathrm{exp}left[icdotmathrm{trace}({mathbf W}{mathbfTheta})right]right} = left|{mathbf I} - 2i{mathbfTheta}{mathbfSigma}right|^{-n/2}

where {mathcal E}(cdot) denotes expectation.


Theorem

If {mathbf W} has a Wishart distribution with m degrees of freedom and variance matrix {mathbf V}---write {mathbf W}sim{mathbf W}_p({mathbf V},m)---and {mathbf C} is a qtimes p matrix of rank q, then In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...

{mathbf C}{mathbf W}{mathbf C'} sim {mathbf W}_qleft({mathbf C}{mathbf V}{mathbf C'},mright)

Corollary 1

If {mathbf z} is a nonzero ptimes 1 constant vector, then {mathbf z'}{mathbf W}{mathbf z}simsigma_z^2chi_m^2.


In this case, chi_m^2 is the chi-square distribution and sigma_z^2={mathbf z'}{mathbf V}{mathbf z} (note that sigma_z^2 is a constant; it is positive because {mathbf V} is positive definite). In probability theory and statistics, the chi-square distribution (also chi-squared or χ2  distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ...


Corollary 2

Consider the case where {mathbf z'}=(0,ldots,0,1,0,ldots,0) (that is, the j-th element is one and all others zero). Then corollary 1 above shows that

w_{jj}simsigma_{jj}chi^2_m

gives the marginal distribution of each of the elements on the matrix's diagonal.


Noted statistician George Seber points out that the Wishart distribution is not called the "multivariate chi-square distribution" because the marginal distribution of the off-diagonal elements is not chi-square. Seber prefers to reserve the term multivariate for the case when all univariate marginals belong to the same family. Multivariate statistics or multivariate statistical analysis in statistics describes a collection of procedures which involve observation and analysis of more than one statistical variable at a time. ...


Estimator of the multivariate normal distribution

The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution. The derivation of the MLE is perhaps surprisingly subtle and elegant. It involves the spectral theorem and the reason why it can be better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices. In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. ...


See also

Image:Bvn-small.png Probability distributions [ view • talk • edit ]
Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular•truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

  Results from FactBites:
 
Wishart distribution - Wikipedia, the free encyclopedia (494 words)
In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables ("random matrices").
Noted statistician George Seber points out that the Wishart distribution is not called the "multivariate chi-square distribution" because the marginal distribution of the off-diagonal elements is not chi-square.
The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution.
  More results at FactBites »

 
 

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