rInvCholWishart {CholWishart}

Cholesky Factor of Random Inverse Wishart Distributed Matrices

Description

Generate n random matrices, distributed according to the Cholesky factor of an inverse Wishart distribution with parameters Sigma and df, W_p(Sigma, df).

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.

Usage

rInvCholWishart(n, df, Sigma)

Arguments

n

integer sample size.

df

numeric parameter, "degrees of freedom".

Sigma

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

Value

a numeric array, say R, of dimension p \times p \times n, where each R[,,i] is a Cholesky decomposition of a realization of the Wishart distribution W_p(Sigma, df). Based on a modification of the existing code for the rWishart function

References

Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Hoboken, N. J.: Wiley Interscience.

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.

See Also

rWishart and rCholWishart

Examples

# How it is parameterized:
set.seed(20180211)
A <- rCholWishart(1L, 10, 3 * diag(5L))[, , 1]
A
set.seed(20180211)
B <- rInvCholWishart(1L, 10, 1 / 3 * diag(5L))[, , 1]
B
crossprod(A) %*% crossprod(B)

set.seed(20180211)
C <- chol(stats::rWishart(1L, 10, 3 * diag(5L))[, , 1])
C