rWishart: Random Wishart Distributed Matrices

Description Usage Arguments Details Value Author(s) References See Also Examples

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.