Wishart distribution information

Wishart
Notation	X ~ Wp(V, n)
Parameters	n > p − 1 degrees of freedom (real); V > 0 scale matrix (p × p pos. def)
Support	X(p × p) positive definite matrix
PDF	Γp is the multivariate gamma function; tr is the trace function;
Mean
Mode	(n − p − 1)V for n ≥ p + 1
Variance
Entropy	see below
CF

In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first formulated the distribution in 1928.^[1] Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).^[2]

It is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector.^[3]

^ Wishart, J. (1928). "The generalised product moment distribution in samples from a normal multivariate population". Biometrika. 20A (1–2): 32–52. doi:10.1093/biomet/20A.1-2.32. JFM 54.0565.02. JSTOR 2331939.
^ Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018), Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (eds.), "Classical Ensembles: Wishart-Laguerre", Introduction to Random Matrices: Theory and Practice, SpringerBriefs in Mathematical Physics, Cham: Springer International Publishing, pp. 89–95, doi:10.1007/978-3-319-70885-0_13, ISBN 978-3-319-70885-0, retrieved 2023-05-17
^ Koop, Gary; Korobilis, Dimitris (2010). "Bayesian Multivariate Time Series Methods for Empirical Macroeconomics". Foundations and Trends in Econometrics. 3 (4): 267–358. doi:10.1561/0800000013.

[Wishart-1] Wishart, J. (1928). "The generalised product moment distribution in samples from a normal multivariate population". Biometrika. 20A (1–2): 32–52. doi:10.1093/biomet/20A.1-2.32. JFM 54.0565.02. JSTOR 2331939.

[2] Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018), Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (eds.), "Classical Ensembles: Wishart-Laguerre", Introduction to Random Matrices: Theory and Practice, SpringerBriefs in Mathematical Physics, Cham: Springer International Publishing, pp. 89–95, doi:10.1007/978-3-319-70885-0_13, ISBN 978-3-319-70885-0, retrieved 2023-05-17

[3] Koop, Gary; Korobilis, Dimitris (2010). "Bayesian Multivariate Time Series Methods for Empirical Macroeconomics". Foundations and Trends in Econometrics. 3 (4): 267–358. doi:10.1561/0800000013.

Wishart distribution information

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Wishart distribution

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List of probability distributions

Probability distribution

Matrix normal distribution

Complex normal distribution

Matrix gamma distribution

Dirichlet distribution

Inverse matrix gamma distribution

Complex inverse Wishart distribution

Wishart

Conjugate prior

Distribution of the product of two random variables

Estimation of covariance matrices

Schur complement

Inverse distribution

Multivariate statistics

Notation	$X ~ W p (V, n)$
Parameters	$n > p - 1$ degrees of freedom (real) $V > 0$ scale matrix ( $p \times p$ pos. def)
Support	$X (p \times p)$ positive definite matrix
PDF	$f_{\mathbf {X} }(\mathbf {X} )={\frac {\|\mathbf {X} \|^{(n-p-1)/2}e^{-\operatorname {tr} (\mathbf {V} ^{-1}\mathbf {X} )/2}}{2^{\frac {np}{2}}\|{\mathbf {V} }\|^{n/2}\Gamma _{p}({\frac {n}{2}})}}$ $Γ p$ is the multivariate gamma function $tr$ is the trace function
Mean	$\operatorname {E} [X]=n{\mathbf {V} }$
Mode	$(n - p - 1) V$ for $n \geq p + 1$
Variance	$\operatorname {Var} (\mathbf {X} _{ij})=n\left(v_{ij}^{2}+v_{ii}v_{jj}\right)$
Entropy	see below
CF	$\Theta \mapsto \left\|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right\|^{-{\frac {n}{2}}}$