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 \(\sum X_i X_i'\) 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.
rPseudoWishart(n, df, Sigma)
| n | integer sample size. |
|---|---|
| df | integer parameter, "degrees of freedom", should be less than the
dimension of |
| Sigma | positive definite \(p \times p\) "scale" matrix, the matrix parameter of the distribution. |
a numeric array, say R, of dimension
\(p \times p \times n\),
where each R[,,i] is a realization of the pseudo Wishart
distribution \(W_p(Sigma, df)\).
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 .
set.seed(20181227) A <- rPseudoWishart(1L, 4L, 5.0 * diag(5L))[, , 1] # A should be singular eigen(A)$values #> [1] 3.632663e+01 1.641923e+01 8.998881e+00 1.990360e+00 1.221245e-15