# rPseudoWishart: Random Pseudo Wishart Distributed Matrices In CholWishart: Cholesky Decomposition of the Wishart Distribution

## Description

Generate n random matrices, distributed according to the pseudo Wishart distribution with parameters `Sigma` and `df`, W_p(Sigma, df), with sample size `df` less than the dimension `p`.

Let X_i, i = 1, 2, ..., df be `df` observations of a multivariate normal distribution with mean 0 and covariance `Sigma`. Then ∑ X_i X_i' is distributed as a pseudo Wishart W_p(Sigma, df). Sometimes this is called a singular Wishart distribution, however, that can be confused with the case where Sigma itself is singular. If cases with a singular Sigma are desired, this function cannot provide them.

## Usage

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

## Arguments

 `n` integer sample size. `df` integer parameter, "degrees of freedom", should be less than the dimension of `p` `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 realization of the pseudo Wishart distribution W_p(Sigma, df).

## References

Diaz-Garcia, Jose A, Ramon Gutierrez Jaimez, and Kanti V Mardia. 1997. “Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory.” Journal of Multivariate Analysis 63 (1): 73–87. doi: 10.1006/jmva.1997.1689.

Uhlig, Harald. "On Singular Wishart and Singular Multivariate Beta Distributions." Ann. Statist. 22 (1994), no. 1, 395–405. doi: 10.1214/aos/1176325375.

`rWishart`, `rInvWishart`, and `rGenInvWishart`
 ```1 2 3 4``` ```set.seed(20181227) A <- rPseudoWishart(1L, 4L, 5.0 * diag(5L))[, , 1] # A should be singular eigen(A)\$values ```