In statistics, the matrix F distribution (or matrix variate F distribution) is a matrix variate generalization of the F distribution which is defined on real-valued positive-definite matrices. In Bayesian statistics it can be used as the semi conjugate prior for the covariance matrix or precision matrix of multivariate normal distributions, and related distributions.[1][2][3][4]

Matrix
Notation
Parameters , scale matrix (pos. def.)
degrees of freedom (real)
degrees of freedom (real)
Support is p × p positive definite matrix
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Mean , for
Variance see below

Density

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The probability density function of the matrix   distribution is:

 

where   and   are   positive definite matrices,   is the determinant, Γp(⋅) is the multivariate gamma function, and   is the p × p identity matrix.

Properties

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Construction of the distribution

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  • The standard matrix F distribution, with an identity scale matrix  , was originally derived by.[1] When considering independent distributions,

  and  , and define  , then  .

  • If   and  , then, after integrating out  ,   has a matrix F-distribution, i.e.,

 
This construction is useful to construct a semi-conjugate prior for a covariance matrix.[3]

  • If   and  , then, after integrating out  ,   has a matrix F-distribution, i.e.,
     
    This construction is useful to construct a semi-conjugate prior for a precision matrix.[4]

Marginal distributions from a matrix F distributed matrix

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Suppose   has a matrix F distribution. Partition the matrices   and   conformably with each other

 

where   and   are   matrices, then we have  .

Moments

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Let  .

The mean is given by:  

The (co)variance of elements of   are given by:[3]

 
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  • The matrix F-distribution has also been termed the multivariate beta II distribution.[5] See also,[6] for a univariate version.
  • A univariate version of the matrix F distribution is the F-distribution. With   (i.e. univariate) and  , and  , the probability density function of the matrix F distribution becomes the univariate (unscaled) F distribution:
     
  • In the univariate case, with   and  , and when setting  , then   follows a half t distribution with scale parameter   and degrees of freedom  . The half t distribution is a common prior for standard deviations[7]

See also

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References

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  1. ^ a b Olkin, Ingram; Rubin, Herman (1964-03-01). "Multivariate Beta Distributions and Independence Properties of the Wishart Distribution". The Annals of Mathematical Statistics. 35 (1): 261–269. doi:10.1214/aoms/1177703748. ISSN 0003-4851.
  2. ^ Dawid, A. P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika. 68 (1): 265–274. doi:10.1093/biomet/68.1.265. ISSN 0006-3444.
  3. ^ a b c Mulder, Joris; Pericchi, Luis Raúl (2018-12-01). "The Matrix-F Prior for Estimating and Testing Covariance Matrices". Bayesian Analysis. 13 (4). doi:10.1214/17-BA1092. ISSN 1936-0975. S2CID 126398943.
  4. ^ a b Williams, Donald R.; Mulder, Joris (2020-12-01). "Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints". Journal of Mathematical Psychology. 99: 102441. doi:10.1016/j.jmp.2020.102441. S2CID 225019695.
  5. ^ Tan, W. Y. (1969-03-01). "Note on the Multivariate and the Generalized Multivariate Beta Distributions". Journal of the American Statistical Association. 64 (325): 230–241. doi:10.1080/01621459.1969.10500966. ISSN 0162-1459.
  6. ^ Pérez, María-Eglée; Pericchi, Luis Raúl; Ramírez, Isabel Cristina (2017-09-01). "The Scaled Beta2 Distribution as a Robust Prior for Scales". Bayesian Analysis. 12 (3). doi:10.1214/16-BA1015. ISSN 1936-0975.
  7. ^ Gelman, Andrew (2006-09-01). "Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper)". Bayesian Analysis. 1 (3). doi:10.1214/06-BA117A. ISSN 1936-0975.