# rWishart: Random Wishart Distributed Matrices

## Description

Generate `n` random matrices, distributed according to the Wishart distribution with parameters `Sigma` and `df`, W_p(Sigma, df).

## Usage

 `1` ```rWishart(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.

## Details

If X1,...,Xm, Xi in R^p is a sample of m independent multivariate Gaussians with mean (vector) 0, and covariance matrix Σ, the distribution of M = X'X is W_p(Σ, m).

Consequently, the expectation of M is

E[M] = m * Sigma.

Further, if `Sigma` is scalar (p = 1), the Wishart distribution is a scaled chi-squared (chi^2) distribution with `df` degrees of freedom, W_1(sigma^2, m) = sigma^2 chi[m]^2.

The component wise variance is

Var(M[i,j]) = m*(S[i,j]^2 + S[i,i] * S[j,j]), where S=Sigma.

## Value

a numeric `array`, say `R`, of dimension p * p * n, where each `R[,,i]` is a positive definite matrix, a realization of the Wishart distribution W_p(Sigma, df).

Douglas Bates

## References

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

`cov`, `rnorm`, `rchisq`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```## Artificial S <- toeplitz((10:1)/10) set.seed(11) R <- rWishart(1000, 20, S) dim(R) # 10 10 1000 mR <- apply(R, 1:2, mean) # ~= E[ Wish(S, 20) ] = 20 * S stopifnot(all.equal(mR, 20*S, tolerance = .009)) ## See Details, the variance is Va <- 20*(S^2 + tcrossprod(diag(S))) vR <- apply(R, 1:2, var) stopifnot(all.equal(vR, Va, tolerance = 1/16)) ```