rInvCholWishart: 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)

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

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.

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. 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

Run this code

# 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