Wishart distribution

Wishart distribution

Probability distribution
name =Wishart
type =density
pdf_

cdf_

parameters =$n > 0!$ deg. of freedom (real)
$mathbf\left\{V\right\} > 0,$ scale matrix ( pos. def)
support =$mathbf\left\{W\right\}!$ is positive definite
pdf =$frac\left\{left|mathbf\left\{W\right\} ight|^frac\left\{n-p-1\right\}\left\{2$
{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 =$n mathbf\left\{V\right\}$
median =
mode =$\left(n-p-1\right)mathbf\left\{V\right\} ext\left\{ for \right\}n geq p+1$
variance =$n\left(v_\left\{ij\right\}^2+v_\left\{ii\right\}v_\left\{jj\right\}\right)$
skewness =
kurtosis =
entropy =
mgf =
char =$Theta mapsto left|\left\{mathbf I\right\} - 2i,\left\{mathbfTheta\right\}\left\{mathbf V\right\} ight|^\left\{-n/2\right\}$

In statistics, the Wishart distribution, named in honor of John Wishart, is a generalization to multiple dimensions of the chi-square distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. It 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.

Definition

Suppose "X" is an "n" &times; "p" matrix, each row of which is independently drawn from "p"-variate normal distribution with zero mean:

:$X_\left\{\left(i\right)\right\}\left\{=\right\}\left(x_i^1,dots,x_i^p\right)^Tsim N_p\left(0,V\right),$

Then the Wishart distribution is the probability distribution of the "p"&times;"p" random matrix

:$S=X^T X = sum_\left\{i = 1\right\}^\left\{n\right\} X_\left\{\left(i\right)\right\} X_\left\{\left(i\right)\right\}^T, ,!$

known as the scatter matrix. One indicates that "S" has that probability distributionby writing

:$Ssim W_p\left(V,n\right).$

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 tests 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" &times; "p" symmetric matrix of random variables that is positive definite. Let V be a (fixed) positive definite matrix of size "p" &times; "p".

Then, if "n" &ge; "p", then W has a Wishart distribution with "n" degrees of freedom if it has a probability density function "f"W given by

:$f_\left\{mathbf W\right\}\left(w\right)=frac\left\{ left|w ight|^\left\{\left(n-p-1\right)/2\right\} expleft \left[ - \left\{ m trace\right\}\left(\left\{mathbf V\right\}^\left\{-1\right\}w/2 \right) ight\right] \right\}\left\{2^\left\{np/2\right\}left|\left\{mathbf V\right\} ight|^\left\{n/2\right\}Gamma_p\left(n/2\right)\right\}$

where &Gamma;"p"(&middot;) is the multivariate gamma function defined as

:$Gamma_p\left(n/2\right)=pi^\left\{p\left(p-1\right)/4\right\}Pi_\left\{j=1\right\}^pGammaleft \left[ \left(n+1-j\right)/2 ight\right] .$

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

Characteristic function

The characteristic function of the Wishart distribution is

:$Theta mapsto left|\left\{mathbf I\right\} - 2i,\left\{mathbfTheta\right\}\left\{mathbf V\right\} ight|^\left\{-n/2\right\}.$

In other words,

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

where $\left\{mathcal E\right\}\left(cdot\right)$ denotes expectation.

(here $Theta$ and $\left\{mathbf I\right\}$ are matrices the same size as $\left\{mathbf V\right\}$ ($\left\{mathbf I\right\}$ is the identity matrix); and $i$ is the square root of minus one).

Theorem

If $scriptstyle \left\{mathbf W\right\}$ has a Wishart distribution with "m" degrees of freedom and variance matrix $scriptstyle \left\{mathbf V\right\}$&mdash;write $scriptstyle \left\{mathbf W\right\}sim\left\{mathbf W\right\}_p\left(\left\{mathbf V\right\},m\right)$&mdash;and $scriptstyle\left\{mathbf C\right\}$ is a "q" &times; "p" matrix of rank "q", then

:$\left\{mathbf C\right\}\left\{mathbf W\right\}\left\{mathbf C\text{'}\right\}sim\left\{mathbf W\right\}_qleft\left(\left\{mathbf C\right\}\left\{mathbf V\right\}\left\{mathbf C\text{'}\right\},m ight\right).$

Corollary 1

If $\left\{mathbf z\right\}$ is a nonzero $p imes 1$ constant vector, then$\left\{mathbf z\text{'}\right\}\left\{mathbf W\right\}\left\{mathbf z\right\}simsigma_z^2chi_m^2$.

In this case, $chi_m^2$ isthe chi-square distribution and $sigma_z^2=\left\{mathbf z\text{'}\right\}\left\{mathbf V\right\}\left\{mathbf z\right\}$ (note that $sigma_z^2$ is a constant; it is positive because $\left\{mathbf V\right\}$ is positive definite).

Corollary 2

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

:$w_\left\{jj\right\}simsigma_\left\{jj\right\}chi^2_m$

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.

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&times;1 matrix than as a mere scalar. See estimation of covariance matrices.

Drawing values from the distribution

The following procedure is due to Smith &amp; Hocking [http://www.jstor.org/pss/2346290] . One can sample random "p" &times; "p" matrices from a "p"-variate Wishart distribution with scale matrix $\left\{ extbf V\right\}$ and "n" degrees of freedom (for $n geq p$) as follows:

# Generate a random "p" &times; "p" lower triangular matrix $\left\{ extbf A\right\}$ such that:
#* $a_\left\{ii\right\}=\left(chi^2_\left\{n-i+1\right\}\right)^\left\{1/2\right\}$, i.e. $a_\left\{ii\right\}$ is the square root of a sample taken from a chi-square distribution $chi^2_\left\{n-i+1\right\}$
#* $a_\left\{ij\right\}$, for
# Compute the Cholesky decomposition of $\left\{ extbf V\right\} = \left\{ extbf L\right\}\left\{ extbf L\right\}^T$.
# Compute the matrix $\left\{ extbf X\right\} = \left\{ extbf L\right\}\left\{ extbf A\right\}\left\{ extbf A\right\}^T\left\{ extbf L\right\}^T$. At this point, $\left\{ extbf X\right\}$ is a sample from the Wishart distribution $W_p\left(\left\{ extbf V\right\},n\right)$.

Note that if $\left\{ extbf V\right\}=\left\{ extbf I\right\}$, the identity matrix, then the sample can be directly obtained from $\left\{ extbf X\right\} = \left\{ extbf A\right\}\left\{ extbf A\right\}^T$ since the Cholesky decomposition of $\left\{ extbf V\right\}=\left\{ extbf I\right\}\left\{ extbf I\right\}^T$.

ee also

*Estimation of covariance matrices
*Hotelling's T-square distribution
*Inverse-Wishart distribution

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