Matrix variate beta distribution

In statistics, the matrix variate beta distribution is a generalization of the beta distribution. It is also called the MANOVA ensemble and the Jacobi ensemble.

If ${\displaystyle U}${\displaystyle U} is a ${\displaystyle p\times p}${\displaystyle p\times p} positive definite matrix with a matrix variate beta distribution, and ${\displaystyle a,b>(p-1)/2}${\displaystyle a,b>(p-1)/2} are real parameters, we write ${\displaystyle U\sim B_{p}\left(a,b\right)}${\displaystyle U\sim B_{p}\left(a,b\right)} (sometimes ${\displaystyle B_{p}^{I}\left(a,b\right)}${\displaystyle B_{p}^{I}\left(a,b\right)}). The probability density function for ${\displaystyle U}${\displaystyle U} is:$${\displaystyle \left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.}$${\displaystyle \left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.}

Here ${\displaystyle \beta _{p}\left(a,b\right)}${\displaystyle \beta _{p}\left(a,b\right)} is the multivariate beta function:

${\displaystyle \beta _{p}\left(a,b\right)={\frac {\Gamma _{p}\left(a\right)\Gamma _{p}\left(b\right)}{\Gamma _{p}\left(a+b\right)}}}${\displaystyle \beta _{p}\left(a,b\right)={\frac {\Gamma _{p}\left(a\right)\Gamma _{p}\left(b\right)}{\Gamma _{p}\left(a+b\right)}}}

where ${\displaystyle \Gamma _{p}\left(a\right)}${\displaystyle \Gamma _{p}\left(a\right)} is the multivariate gamma function given by

${\displaystyle \Gamma _{p}\left(a\right)=\pi ^{p(p-1)/4}\prod _{i=1}^{p}\Gamma \left(a-(i-1)/2\right).}${\displaystyle \Gamma _{p}\left(a\right)=\pi ^{p(p-1)/4}\prod _{i=1}^{p}\Gamma \left(a-(i-1)/2\right).}

If ${\displaystyle U\sim B_{p}(a,b)}${\displaystyle U\sim B_{p}(a,b)} then the density of ${\displaystyle X=U^{-1}}${\displaystyle X=U^{-1}} is given by

${\displaystyle {\frac {1}{\beta _{p}\left(a,b\right)}}\det(X)^{-(a+b)}\det \left(X-I_{p}\right)^{b-(p+1)/2}}${\displaystyle {\frac {1}{\beta _{p}\left(a,b\right)}}\det(X)^{-(a+b)}\det \left(X-I_{p}\right)^{b-(p+1)/2}}

provided that ${\displaystyle X>I_{p}}${\displaystyle X>I_{p}} and ${\displaystyle a,b>(p-1)/2}${\displaystyle a,b>(p-1)/2}.

If ${\displaystyle U\sim B_{p}(a,b)}${\displaystyle U\sim B_{p}(a,b)} and ${\displaystyle H}${\displaystyle H} is a constant ${\displaystyle p\times p}${\displaystyle p\times p} orthogonal matrix, then ${\displaystyle HUH^{T}\sim B(a,b).}${\displaystyle HUH^{T}\sim B(a,b).}

Also, if ${\displaystyle H}${\displaystyle H} is a random orthogonal ${\displaystyle p\times p}${\displaystyle p\times p} matrix which is independent of ${\displaystyle U}${\displaystyle U}, then ${\displaystyle HUH^{T}\sim B_{p}(a,b)}${\displaystyle HUH^{T}\sim B_{p}(a,b)}, distributed independently of ${\displaystyle H}${\displaystyle H}.

If ${\displaystyle A}${\displaystyle A} is any constant ${\displaystyle q\times p}${\displaystyle q\times p}, ${\displaystyle q\leq p}${\displaystyle q\leq p} matrix of rank ${\displaystyle q}${\displaystyle q}, then ${\displaystyle AUA^{T}}${\displaystyle AUA^{T}} has a generalized matrix variate beta distribution, specifically ${\displaystyle AUA^{T}\sim GB_{q}\left(a,b;AA^{T},0\right)}${\displaystyle AUA^{T}\sim GB_{q}\left(a,b;AA^{T},0\right)}.

If ${\displaystyle U\sim B_{p}\left(a,b\right)}${\displaystyle U\sim B_{p}\left(a,b\right)} and we partition ${\displaystyle U}${\displaystyle U} as

${\displaystyle U={\begin{bmatrix}U_{11}&U_{12}\\U_{21}&U_{22}\end{bmatrix}}}${\displaystyle U={\begin{bmatrix}U_{11}&U_{12}\\U_{21}&U_{22}\end{bmatrix}}}

where ${\displaystyle U_{11}}${\displaystyle U_{11}} is ${\displaystyle p_{1}\times p_{1}}${\displaystyle p_{1}\times p_{1}} and ${\displaystyle U_{22}}${\displaystyle U_{22}} is ${\displaystyle p_{2}\times p_{2}}${\displaystyle p_{2}\times p_{2}}, then defining the Schur complement ${\displaystyle U_{22\cdot 1}}${\displaystyle U_{22\cdot 1}} as ${\displaystyle U_{22}-U_{21}{U_{11}}^{-1}U_{12}}${\displaystyle U_{22}-U_{21}{U_{11}}^{-1}U_{12}} gives the following results:

• ${\displaystyle U_{11}}${\displaystyle U_{11}} is independent of ${\displaystyle U_{22\cdot 1}}${\displaystyle U_{22\cdot 1}}
• ${\displaystyle U_{11}\sim B_{p_{1}}\left(a,b\right)}${\displaystyle U_{11}\sim B_{p_{1}}\left(a,b\right)}
• ${\displaystyle U_{22\cdot 1}\sim B_{p_{2}}\left(a-p_{1}/2,b\right)}${\displaystyle U_{22\cdot 1}\sim B_{p_{2}}\left(a-p_{1}/2,b\right)}
• ${\displaystyle U_{21}\mid U_{11},U_{22\cdot 1}}${\displaystyle U_{21}\mid U_{11},U_{22\cdot 1}} has an inverted matrix variate t distribution, specifically ${\displaystyle U_{21}\mid U_{11},U_{22\cdot 1}\sim IT_{p_{2},p_{1}}\left(2b-p+1,0,I_{p_{2}}-U_{22\cdot 1},U_{11}(I_{p_{1}}-U_{11})\right).}${\displaystyle U_{21}\mid U_{11},U_{22\cdot 1}\sim IT_{p_{2},p_{1}}\left(2b-p+1,0,I_{p_{2}}-U_{22\cdot 1},U_{11}(I_{p_{1}}-U_{11})\right).}

Mitra proves the following theorem which illustrates a useful property of the matrix variate beta distribution. Suppose ${\displaystyle S_{1},S_{2}}${\displaystyle S_{1},S_{2}} are independent Wishart ${\displaystyle p\times p}${\displaystyle p\times p} matrices ${\displaystyle S_{1}\sim W_{p}(n_{1},\Sigma ),S_{2}\sim W_{p}(n_{2},\Sigma )}${\displaystyle S_{1}\sim W_{p}(n_{1},\Sigma ),S_{2}\sim W_{p}(n_{2},\Sigma )}. Assume that ${\displaystyle \Sigma }${\displaystyle \Sigma } is positive definite and that ${\displaystyle n_{1}+n_{2}\geq p}${\displaystyle n_{1}+n_{2}\geq p}. If

${\displaystyle U=S^{-1/2}S_{1}\left(S^{-1/2}\right)^{T},}${\displaystyle U=S^{-1/2}S_{1}\left(S^{-1/2}\right)^{T},}

where ${\displaystyle S=S_{1}+S_{2}}${\displaystyle S=S_{1}+S_{2}}, then ${\displaystyle U}${\displaystyle U} has a matrix variate beta distribution ${\displaystyle B_{p}(n_{1}/2,n_{2}/2)}${\displaystyle B_{p}(n_{1}/2,n_{2}/2)}. In particular, ${\displaystyle U}${\displaystyle U} is independent of ${\displaystyle \Sigma }${\displaystyle \Sigma }.

The spectral density is expressed by a Jacobi polynomial.^[1]

The distribution of the largest eigenvalue is well approximated by a transform of the Tracy–Widom distribution.^[2]

• Matrix variate Dirichlet distribution

• Potters, Marc; Bouchaud, Jean-Philippe (2020年11月30日). "7. The Jacobi Ensemble". A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists. Cambridge University Press. doi:10.1017/9781108768900. ISBN 978-1-108-76890-0.
• Forrester, Peter (2010). "3. Laguerre and Jacobi ensembles". Log-gases and random matrices. London Mathematical Society monographs. Princeton: Princeton University Press. ISBN 978-0-691-12829-0.
• Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). "4. Some generalities". An introduction to random matrices. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.
• Mehta, M.L. (2004). "19. Matrix ensembles and classical orthogonal polynomials". Random Matrices. Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7.
• Gupta, A. K.; Nagar, D. K. (1999). Matrix Variate Distributions. Chapman and Hall. ISBN 1-58488-046-5.
• Khatri, C. G. (1992). "Matrix Beta Distribution with Applications to Linear Models, Testing, Skewness and Kurtosis". In Venugopal, N. (ed.). Contributions to Stochastics. John Wiley & Sons. pp. 26–34. ISBN 0-470-22050-3.
• Mitra, S. K. (1970). "A density-free approach to matrix variate beta distribution". The Indian Journal of Statistics. Series A (1961–2002). 32 (1): 81–88. JSTOR 25049638.

• Random matrices
• Multivariate continuous distributions

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