View source for Wishart distribution

{{Probability distribution
 | name       =Wishart
 | type       =density
 | pdf_image  =
 | cdf_image  =
 | notation   ={{math|''X'' ~ ''W<sub>p</sub>''('''V''', ''n'')}}
 | parameters ={{math|''n'' > ''p'' − 1}} [[degrees of freedom (statistics)|degrees of freedom]] ([[real numbers|real]])<br />{{math|'''V''' > 0}} [[scale matrix]] ({{math|''p'' × ''p''}} [[positive definite matrix|pos. def]])
 | support    ={{math|'''X'''(''p'' × ''p'')}}  [[positive definite matrix]]
 | pdf        =<math>f_{\mathbf X}(\mathbf x) = \frac{|\mathbf{x}|^{(n-p-1)/2} e^{-\operatorname{tr}(\mathbf{V}^{-1}\mathbf{x})/2}}{2^\frac{np}{2}|{\mathbf V}|^{n/2}\Gamma_p(\frac n 2)} </math>
*{{math|Γ<sub>''p''</sub>}} is the [[multivariate gamma function]]
*{{math|tr}} is the [[trace (linear algebra)|trace]] function
 | cdf        =
 | mean       =<math>\operatorname{E}[X]=n{\mathbf V}</math>|
 | median     =
 | mode       ={{math|(''n'' − ''p'' − 1)'''V'''}} for {{math|''n'' ≥ ''p'' + 1}}
 | variance   =<math>\operatorname{Var}(\mathbf{X}_{ij}) = n \left (v_{ij}^2+v_{ii}v_{jj} \right )</math>
 | skewness   =
 | kurtosis   =
 | entropy    =[[#Entropy|see below]]
 | mgf        =
 | char       =<math>\Theta \mapsto \left|{\mathbf I} - 2i\,{\mathbf\Theta}{\mathbf V}\right|^{-\frac{n}{2}}</math>
}}

In [[statistics]], the '''Wishart distribution''' is a generalization to multiple dimensions of the [[gamma distribution]]. It is named in honor of [[John Wishart (statistician)|Dr John Wishart]], who first formulated the distribution in 1928.<ref name=Wishart>{{cite journal |first=J. |last=Wishart |authorlink=John Wishart (statistician) |title=The generalised product moment distribution in samples from a normal multivariate population |journal=[[Biometrika]] |volume=20A |issue=1–2 |pages=32–52 |year=1928 |doi=10.1093/biomet/20A.1-2.32 |jfm=54.0565.02 |jstor=2331939}}</ref><ref>econophysics: An Introduction, by S Sinha</ref>

It is a family of [[probability distribution]]s defined over symmetric, [[nonnegative-definite]] [[matrix (math)|matrix]]-valued [[random variable]]s (“random matrices”). These distributions are of great importance in the [[estimation of covariance matrices]] in [[multivariate statistics]]. In [[Bayesian inference|Bayesian statistics]], the Wishart distribution is the [[conjugate prior]] of the [[matrix inverse|inverse]] [[covariance matrix|covariance-matrix]]  of a [[multivariate normal distribution|multivariate-normal random-vector]].<ref>{{cite journal |first=Gary |last=Koop |first2=Dimitris |last2=Korobilis |year=2010 |title=Bayesian Multivariate Time Series Methods for Empirical Macroeconomics |journal=Foundations and Trends in Econometrics |volume=3 |issue=4 |pages=267–358 |doi=10.1561/0800000013 }}</ref>

==Definition==
Suppose {{mvar|G}} is a {{math|''p'' × ''n''}} matrix, each column of which is [[statistical independence|independently]] drawn from a [[multivariate normal distribution|{{mvar|p}}-variate normal distribution]] with zero mean:

:<math>G_{(i)} = (g_i^1,\dots,g_i^p)^T\sim N_p(0,V).</math>

Then the Wishart distribution is the [[probability distribution]] of the {{math|''p'' × ''p''}} random matrix <ref>{{cite book |first=A. K. |last=Gupta |first2=D. K. |last2=Nagar |date=2000 |title=Matrix Variate Distributions |publisher=Chapman & Hall /CRC |isbn=1584880465}}</ref>

:<math>S= G G^T = \sum_{i=1}^n G_{(i)}G_{(i)}^T</math>

known as the [[scatter matrix]]. One indicates that {{mvar|S}} has that probability distribution by writing

:<math>S\sim W_p(V,n).</math>

The positive integer {{mvar|n}} is the number of ''[[degrees of freedom (statistics)|degrees of freedom]]''.  Sometimes this is written {{math|''W''(''V'', ''p'', ''n'')}}. For {{math|''n'' ≥ ''p''}} the matrix {{mvar|S}} is invertible with probability {{math|1}} if {{mvar|V}} is invertible.

If {{math|''p'' {{=}} ''V'' {{=}} 1}} then this distribution is a [[chi-squared distribution]] with {{mvar|n}} degrees of freedom.

==Occurrence==
The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a [[multivariate normal distribution]].  It occurs frequently in [[likelihood-ratio test]]s in multivariate statistical analysis.  It also arises in the spectral theory of [[Random matrix|random matrices]]{{Citation needed|date=October 2010}} and in multidimensional Bayesian analysis.<ref>{{cite book |last=Gelman |first=Andrew |date=2003 |title=Bayesian Data Analysis |url=http://www.stat.columbia.edu/~gelman/book/ |publisher=Chapman & Hall |page=582 |isbn=158488388X |access-date=3 June 2015 |location=Boca Raton, Fla. |edition=2nd}}</ref> It is also encountered in wireless communications, while analyzing the performance of [[Rayleigh fading]] [[MIMO]] wireless channels .<ref>{{cite journal| last=Zanella| first=A.|author2=Chiani, M. |author3=Win, M.Z. |title=On the marginal distribution of the eigenvalues of wishart matrices| journal=IEEE Transactions on Communications|date=April 2009| volume=57| issue=4| pages=1050–1060 | doi=10.1109/TCOMM.2009.04.070143| url=https://dspace.mit.edu/bitstream/1721.1/66900/1/Zanella-2009-On%20the%20Marginal%20Distribution%20of%20the%20Eigenvalues%20of%20Wishart%20Matrices.pdf}}</ref>

==Probability density function==
The Wishart distribution can be [[characterization (mathematics)|characterized]] by its [[probability density function]] as follows:

Let {{math|'''X'''}} be a {{math|''p'' × ''p''}} symmetric matrix of random variables that is [[Positive-definite matrix|positive definite]]. Let {{math|'''V'''}} be a (fixed) symmetric positive definite matrix of size {{math|''p'' × ''p''}}.

Then, if {{math|''n'' ≥ ''p''}}, {{math|'''X'''}} has a Wishart distribution with {{mvar|n}} degrees of freedom if it has the [[probability density function]]

:<math> f_{\mathbf X} (\mathbf x) = \frac{1}{2^{np/2} \left|{\mathbf V}\right|^{n/2} \Gamma_p\left(\frac {n}{2}\right ) }{\left|\mathbf{x}\right|}^{(n-p-1)/2} e^{-(1/2)\operatorname{tr}({\mathbf V}^{-1}\mathbf{x})}</math>

where <math>\left|{\mathbf x}\right|</math> is the [[determinant]] of <math>\mathbf x</math> and {{math|Γ<sub>''p''</sub>}} is the [[multivariate gamma function]] defined as

:<math>\Gamma_p \left (\frac n 2 \right )= \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma\left( \frac{n}{2} - \frac{j-1}{2} \right ).</math>

The density above is not the joint density of all the <math>p^2</math> elements of the random matrix {{math|'''X'''}} (such {{nowrap|<math>p^2</math>-dimensional}} density does not exists because of the symmetry constrains <math>X_{ij}=X_{ji}</math>), it is rather the joint density of <math>p(p+1)/2</math> elements <math>X_{ij}</math> for <math>i\le j</math> (<ref name=Wishart />, page 38). Also, the density formula above applies only to positive definite matrices <math>\mathbf x;</math> for other matrices the density is equal to zero.

The joint-eigenvalue density for the eigenvalues <math>\lambda_1,\dots \lambda_p\ge 0</math> of a random matrix <math> \mathbf{X}\sim W_p(\mathbf{I},n)</math> is <ref>{{cite book |last=Muirhead |first=Robb J. |date=2005 |title=Aspects of Multivariate Statistical Theory |publisher=Wiley Interscience |isbn=0471769851 |edition=2nd}}</ref>, <ref name="Anderson"/>

: <math>c_{n,p}e^{-\frac{1}{2}\sum_i\lambda_i}\prod \lambda_i^{(n-p-1)/2}\prod_{i<j}|\lambda_i-\lambda_j|</math>

where <math>c_{n,p}</math>is a constant.

In fact the above definition can be extended to any real {{math|''n'' > ''p'' − 1}}.  If {{math|''n'' ≤ ''p'' − 1}}, then the Wishart no longer has a density&mdash;instead it represents a singular distribution that takes values in a lower-dimension subspace of the space of {{math|''p'' × ''p''}} matrices.<ref name="Uhlig1994">{{Cite journal | doi = 10.1214/aos/1176325375| title = On Singular Wishart and Singular Multivariate Beta Distributions| journal = The Annals of Statistics| volume = 22| pages = 395–405| year = 1994| last1 = Uhlig | first1 = H. }}</ref>

==Use in Bayesian statistics ==
In [[Bayesian statistics]], in the context of the [[multivariate normal distribution]], the Wishart distribution is the conjugate prior to the precision matrix {{math|'''Ω''' {{=}} '''Σ'''<sup>−1</sup>}}, where {{math|'''Σ'''}} is the covariance matrix.<ref name="bishop"/>{{rp|135}}

=== Choice of parameters ===
The least informative, proper Wishart prior is obtained by setting {{math|''n'' {{=}} ''p''}}.{{Citation needed|date=June 2014}}

The prior mean of {{math|''W<sub>p</sub>''('''V''', ''n'')}} is {{math|''n'''''V'''}}, suggesting that a reasonable choice for {{math|'''V'''}} would be {{math|''n''<sup>−1</sup>'''Σ'''<sub>0</sub>}}, where {{math|'''Σ'''<sub>0</sub>}} is some prior guess for the covariance matrix.

==Properties==

===Log-expectation===
The following formula plays a role in [[variational Bayes]] derivations for [[Bayes network]]s
involving the Wishart distribution: <ref name="bishop"/>{{rp|693}}

:<math>\operatorname{E}[\, \ln\left|\mathbf{X}\right|\, ] =  \psi_p\left(\frac n 2\right) + p \, \ln(2) + \ln|\mathbf{V}|</math>

where <math>\psi_p</math> is the multivariate digamma function (the derivative of the log of the [[multivariate gamma function]]).

===Log-variance===
The following variance computation could be of help in Bayesian statistics:

: <math>\operatorname{Var}\left[\, \ln\left|\mathbf{X}\right| \,\right]=\sum_{i=1}^p \psi_1\left(\frac{n+1-i} 2\right)</math>

where <math>\psi_1</math> is the trigamma function. This comes up when computing the Fisher information of the Wishart random variable.

===Entropy===
The [[information entropy]] of the distribution has the following formula:<ref name="bishop"/>{{rp|693}}

:<math>\operatorname{H}\left[\, \mathbf{X} \,\right] = -\ln \left( B(\mathbf{V},n) \right) -\frac{n-p-1}{2} \operatorname{E}\left[\, \ln\left|\mathbf{X}\right|\,\right] + \frac{np}{2}</math>

where {{math|''B''('''V''', ''n'')}} is the [[normalizing constant]] of the distribution:

:<math>B(\mathbf{V},n) = \frac{1}{\left|\mathbf{V}\right|^{n/2} 2^{np/2}\Gamma_p\left(\frac n 2 \right)}.</math>

This can be expanded as follows:

:<math>
\begin{align}
\operatorname{H}\left[\, \mathbf{X}\, \right] & = \frac{n}{2} \ln \left|\mathbf{V}\right| +\frac{n p}{2} \ln 2 + \ln \Gamma_p \left(\frac{n}{2} \right) - \frac{n-p-1}{2} \operatorname{E}\left[\, \ln\left|\mathbf{X}\right|\, \right] + \frac{n p}{2} \\[8pt]
&= \frac{n}{2} \ln\left|\mathbf{V}\right| + \frac{n p}{2} \ln 2 + \ln\Gamma_p\left(\frac{n}{2} \right) - \frac{n-p-1} 2 \left( \psi_p \left(\frac{n}{2}\right) + p\ln 2 + \ln\left|\mathbf{V}\right|\right) + \frac{n p}{2} \\[8pt]
&= \frac{n}{2} \ln\left|\mathbf{V}\right| + \frac{n p}{2} \ln 2 + \ln\Gamma_p\left(\frac n 2\right)
 - \frac{n-p-1}{2} \psi_p\left(\frac n 2 \right) - \frac{n-p-1} 2 \left(p\ln 2 +\ln\left|\mathbf{V}\right| \right) + \frac{n p}{2} \\[8pt]
&= \frac{p+1}{2} \ln\left|\mathbf{V}\right| + \frac1 2 p(p+1) \ln 2 + \ln\Gamma_p\left(\frac n 2\right) - \frac{n-p-1}{2} \psi_p\left(\frac n 2 \right)  + \frac{n p}{2}
\end{align}
</math>

===Cross-entropy===
The [[cross entropy]] of two Wishart distributions <math>p_0</math> with parameters <math>n_0, V_0</math> and <math>p_1</math> with parameters <math>n_1, V_1</math> is

:<math>\begin{align}
H(p_0, p_1) &= \operatorname{E}_{p_0}[\, -\log p_1\, ]\\[8pt]
&= \operatorname{E}_{p_0} \left[\, -\log \frac{\left|\mathbf{X}\right|^{(n_1 - p - 1)/2} e^{-\operatorname{tr}(\mathbf{V}_1^{-1} \mathbf{X})/2}}{2^{n_1 p/2} \left|\mathbf{V}_1\right|^{n_1/2} \Gamma_p\left(\tfrac{n_1}{2}\right)} \right]\\[8pt]
&= \tfrac{n_1 p} 2 \log 2 + \tfrac{n_1} 2 \log \left|\mathbf{V}_1\right| + \log \Gamma_p(\tfrac{n_1} 2) - \tfrac{n_1 - p - 1} 2 \operatorname{E}_{p_0}\left[\, \log\left|\mathbf{X}\right|\, \right] + \tfrac{1}{2}\operatorname{E}_{p_0}\left[\, \operatorname{tr}\left(\,\mathbf{V}_1^{-1}\mathbf{X}\,\right) \,  \right] \\[8pt]
&= \tfrac{n_1 p}{2} \log 2 + \tfrac{n_1} 2 \log \left|\mathbf{V}_1\right| + \log \Gamma_p(\tfrac{n_1}{2}) - \tfrac{n_1 - p - 1}{2} \left( \psi_p(\tfrac{n_0} 2) + p \log 2 + \log \left|\mathbf{V}_0\right|\right)+ \tfrac{1}{2} \operatorname{tr}\left(\, \mathbf{V}_1^{-1} n_0 \mathbf{V}_0\, \right) \\[8pt]
&=-\tfrac{n_1}{2} \log \left|\, \mathbf{V}_1^{-1} \mathbf{V}_0\, \right| +  \tfrac{p+1} 2 \log \left|\mathbf{V}_0\right| + \tfrac{n_0} 2 \operatorname{tr}\left(\, \mathbf{V}_1^{-1} \mathbf{V}_0\right)+ \log \Gamma_p\left(\tfrac{n_1}{2}\right) - \tfrac{n_1 - p - 1}{2} \psi_p(\tfrac{n_0}{2})  +  \tfrac{p(p+1)}{2} \log 2
\end{align}</math>

Note that when <math>p_0=p_1</math> we recover the entropy.

===KL-divergence===
The [[Kullback–Leibler divergence]] of <math>p_1</math> from <math>p_0</math> is

:<math>
\begin{align}
D_{KL}(p_0 \| p_1) & = H(p_0, p_1) - H(p_0) \\[6pt]
 & =-\frac{n_1} 2 \log |\mathbf{V}_1^{-1} \mathbf{V}_0|  + \frac{n_0}{2}(\operatorname{tr}(\mathbf{V}_1^{-1} \mathbf{V}_0) - p)+ \log \frac{\Gamma_p\left(\frac{n_1} 2 \right)}{\Gamma_p\left(\frac{n_0} 2 \right)} + \tfrac{n_0 - n_1 } 2 \psi_p\left(\frac{n_0} 2\right)
\end{align}
</math>

===Characteristic function===
The [[characteristic function (probability theory)|characteristic function]] of the Wishart distribution is

:<math>\Theta \mapsto \left|\, {\mathbf I} - 2i\,{\mathbf\Theta}\,{\mathbf V}\, \right|^{-n/2}.</math>

In other words,

:<math>\Theta \mapsto \operatorname{E}\left[ \, \exp\left( \,i \operatorname{tr}\left(\,\mathbf{X}{\mathbf\Theta}\,\right)\,\right)\, \right] = \left|\, {\mathbf I} - 2i\, {\mathbf\Theta}\,{\mathbf V}\, \right|^{-n/2} </math>

where {{math|E[⋅]}} denotes expectation. (Here {{math|Θ}} and {{math|'''I'''}} are matrices the same size as {{math|'''V'''}}({{math|'''I'''}} is the [[identity matrix]]); and {{mvar|i}} is the square root of&nbsp;−1).<ref name="Anderson">{{cite book | last = Anderson | first = T. W. | authorlink = T. W. Anderson | title = An Introduction to Multivariate Statistical Analysis | publisher = [[Wiley Interscience]] | edition = 3rd
 | location = Hoboken, N. J. | year = 2003 | page = 259 | isbn = 0-471-36091-0 }}</ref>

Since the determinant's range contains a closed line through the origin for matrix dimensions greater than two, the above formula is only correct for small values of the Fourier variable. (see https://arxiv.org/pdf/1901.09347.pdf)

==Theorem==
If a {{math|''p'' × ''p''}} random matrix {{math|'''X'''}} has a Wishart distribution with {{mvar|m}} degrees of freedom and variance matrix {{math|'''V'''}} — write <math>\mathbf{X}\sim\mathcal{W}_p({\mathbf V},m)</math> — and {{math|'''C'''}} is a {{math|''q'' × ''p''}} matrix of [[rank (matrix theory)|rank]] {{mvar|q}}, then <ref name="rao">{{cite book |last=Rao |first=C. R. |title=Linear Statistical Inference and its Applications |location= |publisher=Wiley |year=1965 |page=535 }}</ref>

:<math>\mathbf{C}\mathbf{X}{\mathbf C}^T \sim \mathcal{W}_q\left({\mathbf C}{\mathbf V}{\mathbf C}^T,m\right).</math>

===Corollary 1===
If {{math|'''z'''}} is a nonzero {{math|''p'' × 1}} constant vector, then:<ref name="rao"/>

:<math>{\mathbf z}^T\mathbf{X}{\mathbf z}\sim\sigma_z^2\chi_m^2.</math>

In this case, <math>\chi_m^2</math> is the [[chi-squared distribution]] and <math>\sigma_z^2={\mathbf z}^T{\mathbf V}{\mathbf z}</math> (note that <math>\sigma_z^2</math> is a constant; it is positive because {{math|'''V'''}} is positive definite).

===Corollary 2===
Consider the case where {{math|'''z'''<sup>''T''</sup> {{=}} (0, ..., 0, 1, 0, ..., 0)}} (that is, the {{mvar|j}}-th element is one and all others zero). Then corollary 1 above shows that

:<math>w_{jj}\sim\sigma_{jj}\chi^2_m</math>

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

[[George Seber]] points out that the Wishart distribution is not called the “multivariate chi-squared distribution” because the marginal distribution of the [[off-diagonal element]]s is not chi-squared. Seber prefers to reserve the term [[multivariate statistics|multivariate]] for the case when all univariate marginals belong to the same family.<ref>{{cite book | last = Seber | first = George A. F. | title = Multivariate Observations | publisher = [[John Wiley & Sons|Wiley]] | year = 2004 | isbn = 978-0471691211 }}</ref>

==Estimator of the multivariate normal distribution==
The Wishart distribution is the [[sampling distribution]] of the [[maximum likelihood|maximum-likelihood estimator]] (MLE) of the [[covariance matrix]] of a [[multivariate normal distribution]].<ref>{{cite book |first=C. |last=Chatfield |first2=A. J. |last2=Collins |year=1980 |title=Introduction to Multivariate Analysis |location=London |publisher=Chapman and Hall |pages=103–108 |isbn=0-412-16030-7 }}</ref> A [[estimation of covariance matrices|derivation of the MLE]] uses the [[spectral theorem]].

==Bartlett decomposition==
The '''Bartlett decomposition''' of a matrix {{math|'''X'''}} from a {{mvar|p}}-variate Wishart distribution with scale matrix {{math|'''V'''}} and {{mvar|n}} degrees of freedom is the factorization:

:<math>\mathbf{X} = {\textbf L}{\textbf A}{\textbf A}^T{\textbf L}^T,</math>

where {{math|'''L'''}} is the [[Cholesky decomposition|Cholesky factor]] of {{math|'''V'''}}, and:

:<math>\mathbf A = \begin{pmatrix}
c_1 & 0 & 0 & \cdots & 0\\
n_{21} & c_2 &0 & \cdots& 0 \\
n_{31} & n_{32} & c_3 & \cdots & 0\\
\vdots & \vdots & \vdots &\ddots & \vdots \\
n_{p1} & n_{p2} & n_{p3} &\cdots & c_p
\end{pmatrix}</math>

where <math>c_i^2 \sim \chi^2_{n-i+1}</math> and {{math|''n<sub>ij</sub>'' ~ ''N''(0, 1)}} independently.<ref>{{cite book | last = Anderson | first = T. W. | authorlink = T. W. Anderson | title = An Introduction to Multivariate Statistical Analysis | publisher = [[Wiley Interscience]] | edition = 3rd | location = Hoboken, N. J. | year = 2003 | page = 257 | isbn = 0-471-36091-0 }}</ref> This provides a useful method for obtaining random samples from a Wishart distribution.<ref>{{cite journal |title=Algorithm AS 53: Wishart Variate Generator |first1= W. B. |last1=Smith |first2= R. R. |last2=Hocking |journal=[[Journal of the Royal Statistical Society, Series C]] |volume=21 |issue=3 |year=1972 |pages=341&ndash;345 |jstor=2346290}}</ref>

==Marginal distribution of matrix elements==
Let {{math|'''V'''}} be a {{math|2 × 2}} variance matrix characterized by [[Pearson product-moment correlation coefficient|correlation coefficient]] {{math|−1 < ''ρ'' < 1}} and {{math|'''L'''}} its lower Cholesky factor:

:<math>\mathbf{V} = \begin{pmatrix}
\sigma_1^2 & \rho \sigma_1 \sigma_2 \\
\rho \sigma_1 \sigma_2 & \sigma_2^2
\end{pmatrix},
\qquad
\mathbf{L} = \begin{pmatrix}
\sigma_1 & 0 \\
\rho \sigma_2 & \sqrt{1-\rho^2} \sigma_2
\end{pmatrix}</math>

Multiplying through the Bartlett decomposition above, we find that a random sample from the {{math|2 × 2}} Wishart distribution is

:<math>\mathbf{X} = \begin{pmatrix}
\sigma_1^2 c_1^2 & \sigma_1 \sigma_2 \left (\rho c_1^2 + \sqrt{1-\rho^2} c_1 n_{21} \right ) \\
\sigma_1 \sigma_2 \left (\rho c_1^2 + \sqrt{1-\rho^2} c_1 n_{21} \right ) & \sigma_2^2 \left(\left (1-\rho^2 \right ) c_2^2 + \left (\sqrt{1-\rho^2} n_{21} + \rho c_1 \right )^2 \right)
\end{pmatrix}</math>

The diagonal elements, most evidently in the first element, follow the {{math|''χ''<sup>2</sup>}} distribution with {{mvar|n}} degrees of freedom (scaled by {{math|''σ''<sup>2</sup>}}) as expected.  The off-diagonal element is less familiar but can be identified as a [[normal variance-mean mixture]] where the mixing density is a {{math|''χ''<sup>2</sup>}} distribution.  The corresponding marginal probability density for the off-diagonal element is therefore the [[variance-gamma distribution]]

:<math>f(x_{12}) =  \frac{\left | x_{12} \right |^{\frac{n-1}{2}}}{\Gamma\left(\frac{n}{2}\right) \sqrt{2^{n-1} \pi \left (1-\rho^2 \right ) \left (\sigma_1 \sigma_2 \right )^{n+1}}} \cdot K_{\frac{n-1}{2}} \left(\frac{\left |x_{12} \right |}{\sigma_1 \sigma_2 \left (1-\rho^2 \right )}\right) \exp{\left(\frac{\rho x_{12}}{\sigma_1 \sigma_2 (1-\rho^2)}\right)}</math>

where {{math|''K<sub>ν</sub>''(''z'')}} is the [[modified Bessel function of the second kind]].<ref>{{cite journal | last1 = Pearson | first1 = Karl | author1-link = Karl Pearson | last2 = Jeffery | first2 = G. B. | author2-link = George Barker Jeffery | last3 = Elderton | first3 = Ethel M. | author3-link = Ethel M. Elderton | title = On the Distribution of the First Product Moment-Coefficient, in Samples Drawn from an Indefinitely Large Normal Population | journal = Biometrika | volume = 21 | issue = | pages = 164–201 | publisher = Biometrika Trust | date = December 1929  | jstor = 2332556 | doi = 10.2307/2332556}}</ref> Similar results may be found for higher dimensions, but the interdependence of the off-diagonal correlations becomes increasingly complicated.  It is also possible to write down the [[moment-generating function]] even in the ''noncentral'' case (essentially the ''n''th power of Craig (1936)<ref>{{cite journal | last = Craig | first = Cecil C. | title = On the Frequency Function of xy | journal = Ann. Math. Statist. | volume = 7 | issue = | pages = 1–15 | year = 1936 | url = http://projecteuclid.org/euclid.aoms/1177732541 | doi = 10.1214/aoms/1177732541}}</ref> equation 10) although the probability density becomes an infinite sum of Bessel functions.

==The range of the shape parameter==
It can be shown <ref>{{cite journal |doi=10.1214/aop/1176990455 |last=Peddada and Richards |first1=Shyamal Das |last2=Richards |first2=Donald St. P. |title=Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution, |journal=[[Annals of Probability]] |volume=19 |issue=2 |pages=868&ndash;874 |year=1991 }}</ref> that the Wishart distribution can be defined if and only if the shape parameter {{math|'''n'''}} belongs to the set

:<math>\Lambda_p:=\{0,\ldots,p-1\}\cup \left(p-1,\infty\right).</math>

This set is named after Gindikin, who introduced it<ref>{{cite journal |doi=10.1007/BF01078179 |first=S.G. |last=Gindikin |title=Invariant generalized functions in homogeneous domains, |journal=[[Funct. Anal. Appl.]] |volume=9 |issue=1 |pages=50&ndash;52 |year=1975}}</ref> in the seventies in the context of gamma distributions on homogeneous cones. However, for the new parameters in the discrete spectrum of the Gindikin ensemble, namely,

:<math>\Lambda_p^*:=\{0, \ldots, p-1\},</math>

the corresponding Wishart distribution has no Lebesgue density.

== Relationships to other distributions ==
*The Wishart distribution is related to the [[inverse-Wishart distribution]], denoted by <math>W_p^{-1}</math>, as follows: If {{math|'''X''' ~ ''W<sub>p</sub>''('''V''', ''n'')}} and if we do the change of variables {{math|'''C''' {{=}} '''X'''<sup>−1</sup>}}, then <math>\mathbf{C}\sim W_p^{-1}(\mathbf{V}^{-1},n)</math>. This relationship may be derived by noting that the absolute value of the [[Jacobian determinant]] of this change of variables is {{math|{{!}}'''C'''{{!}}<sup>''p''+1</sup>}}, see for example equation (15.15) in.<ref>{{cite journal |first=Paul S. |last=Dwyer |title=Some Applications of Matrix Derivatives in Multivariate Analysis |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |year=1967 |volume=62 |issue=318 |pages=607–625 |jstor=2283988 }}</ref>
* In [[Bayesian statistics]], the Wishart distribution is a [[conjugate prior]] for the [[Precision (statistics)|precision parameter]] of the [[multivariate normal distribution]], when the mean parameter is known.<ref name = "bishop">{{cite book |first=C. M. |last=Bishop |title=Pattern Recognition and Machine Learning |location= |publisher=Springer |year=2006 }}</ref>
* A generalization is the [[multivariate gamma distribution]].
* A different type of generalization is the [[normal-Wishart distribution]], essentially the product of a [[multivariate normal distribution]] with a Wishart distribution.

==See also==
{{Colbegin}}
* [[Chi-squared distribution]]
* [[Complex Wishart distribution]]
* [[F-distribution]]
* [[Gamma distribution]]
* [[Hotelling's T-squared distribution]]
* [[Inverse-Wishart distribution]]
* [[Multivariate gamma distribution]]
* [[Student's t-distribution]]
* [[Wilks' lambda distribution]]
{{Colend}}

==References==
{{reflist|30em}}

==External links==
* [https://github.com/zweng/rmg A C++ library for random matrix generator]

{{ProbDistributions|multivariate}}

{{DEFAULTSORT:Wishart Distribution}}
[[Category:Continuous distributions]]
[[Category:Multivariate continuous distributions]]
[[Category:Covariance and correlation]]
[[Category:Random matrices]]
[[Category:Conjugate prior distributions]]
[[Category:Exponential family distributions]]