CholWishart (version 1.1.4)

dWishart: 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)

Value

Density or log of density

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.

Functions

• dInvWishart(): density for the inverse Wishart distribution.

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 \(\delta = \nu - p + 1\).

References

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