# rInvCholWishart: Cholesky Factor of Random Inverse Wishart Distributed... In CholWishart: Cholesky Decomposition of the Wishart Distribution

## 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 ~ IW_p(Sigma, df) then X^{-1} ~ W_p(Sigma^{-1}, df). Dawid (1981) has a different definition: if X ~ W_p(Sigma^{-1}, df) and df > p - 1, then X^{-1} = Y ~ IW(Sigma, delta), where delta = df - p + 1.

## Usage

 `1` ```rInvCholWishart(n, df, Sigma) ```

## Arguments

 `n` integer sample size. `df` numeric parameter, "degrees of freedom". `Sigma` positive definite (p * p) "scale" matrix, the matrix parameter of the distribution.

## Value

a numeric array, say `R`, of dimension p * p * 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.

`rWishart` and `rCholWishart`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```# 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 ```