- Wishart distribution
Probability distribution
name =Wishart
type =density
pdf_
cdf_
parameters = deg. of freedom (real)
scale matrix ( pos. def)
support = is positive definite
pdf = {2^frac{np}{2}left|{mathbf V} ight|^frac{n}{2}Gamma_p(frac{n}{2})} expleft(-frac{1}{2}{ m Tr}({mathbf V}^{-1}mathbf{W}) ight)
cdf =
mean =
median =
mode =
variance =
skewness =
kurtosis =
entropy =
mgf =
char =In
statistics , the Wishart distribution, named in honor of John Wishart, is a generalization to multiple dimensions of thechi-square distribution , or, in the case of non-integer degrees of freedom, of thegamma distribution . It is any of a family ofprobability distribution s for nonnegative-definite matrix-valuedrandom variable s ("random matrices"). These distributions are of great importance in theestimation of covariance matrices inmultivariate statistics .Definition
Suppose "X" is an "n" × "p" matrix, each row of which is independently drawn from "p"-variate normal distribution with zero mean:
:
Then the Wishart distribution is the
probability distribution of the "p"×"p" random matrix:
known as the
scatter matrix . One indicates that "S" has that probability distributionby writing:
The positive integer "n" is the number of "degrees of freedom". Sometimes this is written "W"("V", "p", "n").
If "p" = 1 and "V" = 1 then this distribution is a
chi-square distribution with "n" degrees of freedom.Occurrence
The Wishart distribution arises frequently in
likelihood-ratio test s in multivariate statistical analysis.It also arises in the spectral theory of random matrices.Probability density function
The Wishart distribution can be characterized by its
probability density function , as follows.Let W be a "p" × "p" symmetric matrix of random variables that is
positive definite . Let V be a (fixed) positive definite matrix of size "p" × "p".Then, if "n" ≥ "p", then W has a Wishart distribution with "n" degrees of freedom if it has a
probability density function "f"W given by:
where Γ"p"(·) is the
multivariate gamma function defined as:
In fact the above definition can be extended to any real "n" > "p" − 1.
Characteristic function
The characteristic function of the Wishart distribution is
:
In other words,
:
where denotes expectation.
(here and are matrices the same size as ( is the
identity matrix ); and is the square root of minus one).Theorem
If has a Wishart distribution with "m" degrees of freedom and variance matrix —write —and is a "q" × "p" matrix of rank "q", then
:
Corollary 1
If is a nonzero constant vector, then.
In this case, isthe
chi-square distribution and (note that is a constant; it is positive because is positive definite).Corollary 2
Consider the case where (that is, the "j"th element is one and all others zero). Then corollary 1 above shows that
:
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.Estimator of the multivariate normal distribution
The Wishart distribution is the
probability distribution of the maximum-likelihood estimator (MLE) of thecovariance matrix of amultivariate normal distribution . The derivation of the MLE is perhaps surprisingly subtle and elegant. It involves thespectral 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. Seeestimation of covariance matrices .Drawing values from the distribution
The following procedure is due to Smith & Hocking [http://www.jstor.org/pss/2346290] . One can sample random "p" × "p" matrices from a "p"-variate Wishart distribution with scale matrix and "n" degrees of freedom (for ) as follows:
# Generate a random "p" × "p" lower
triangular matrix such that:
#* , i.e. is the square root of a sample taken from achi-square distribution
#* , for
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