Inverse matrix gamma distribution

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Inverse matrix gamma
Notation	${\rm {IMG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Psi }})$ {\displaystyle {\rm {IMG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Psi }})}
Parameters	$\alpha >(p-1)/2$ {\displaystyle \alpha >(p-1)/2} shape parameter $\beta >0$ {\displaystyle \beta >0} scale parameter ${\boldsymbol {\Psi }}$ {\displaystyle {\boldsymbol {\Psi }}} scale (positive-definite real $p\times p$ {\displaystyle p\times p} matrix)
Support	$\mathbf {X}$ {\displaystyle \mathbf {X} } positive-definite real $p\times p$ {\displaystyle p\times p} matrix
PDF	${\frac {\|{\boldsymbol {\Psi }}\|^{\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}\|\mathbf {X} \|^{-\alpha -(p+1)/2}\exp \left(-{\frac {1}{\beta }}{\rm {tr}}\left({\boldsymbol {\Psi }}\mathbf {X} ^{-1}\right)\right)$ {\displaystyle {\frac {\|{\boldsymbol {\Psi }}\|^{\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}\|\mathbf {X} \|^{-\alpha -(p+1)/2}\exp \left(-{\frac {1}{\beta }}{\rm {tr}}\left({\boldsymbol {\Psi }}\mathbf {X} ^{-1}\right)\right)} $\Gamma _{p}$ {\displaystyle \Gamma _{p}} is the multivariate gamma function.

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices.^[1] It is a more general version of the inverse Wishart distribution, and is used similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. The compound distribution resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance matrix is a generalized matrix t-distribution.^{[citation needed ]}

This reduces to the inverse Wishart distribution with $\nu$ {\displaystyle \nu } degrees of freedom when $\beta =2,\alpha ={\frac {\nu }{2}}$ {\displaystyle \beta =2,\alpha ={\frac {\nu }{2}}}.

References

[edit ]

^ Iranmanesha, Anis; Arashib, M.; Tabatabaeya, S. M. M. (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics. 5 (2): 33–43.

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Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda