dWishart {CholWishart}

Density for Random Wishart Distributed Matrices

Description

Compute the density of an observation of a random Wishart distributed matrix (dWishart) or an observation from the inverse Wishart distribution (dInvWishart).

Usage

dWishart(x, df, Sigma, log = TRUE)

dInvWishart(x, df, Sigma, log = TRUE)

Arguments

x

positive definite p \times p observations for density estimation - either one matrix or a 3-D array.

df

numeric parameter, "degrees of freedom".

Sigma

positive definite p \times p "scale" matrix, the matrix parameter of the distribution.

log

logical, whether to return value on the log scale.

Details

Note there are different ways of parameterizing the Inverse Wishart distribution, so check which one you need. Here, If X \sim IW_p(\Sigma, \nu) then X^{-1} \sim W_p(\Sigma^{-1}, \nu). Dawid (1981) has a different definition: if X \sim W_p(\Sigma^{-1}, \nu) and \nu > p - 1, then X^{-1} = Y \sim IW(\Sigma, \delta), where \delta = \nu - p + 1.

Value

Density or log of density

Functions

• dInvWishart: density for the inverse Wishart distribution.

References

Dawid, A. (1981). Some Matrix-Variate Distribution Theory: Notational Considerations and a Bayesian Application. Biometrika, 68(1), 265-274. doi: 10.2307/2335827

Gupta, A. K. and D. K. Nagar (1999). Matrix variate distributions. Chapman and Hall.

Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.

Examples

set.seed(20180222)
A <- rWishart(1, 10, diag(4))[, , 1]
A
dWishart(x = A, df = 10, Sigma = diag(4L), log = TRUE)
dInvWishart(x = solve(A), df = 10, Sigma = diag(4L), log = TRUE)