rGenInvWishart {CholWishart} | R Documentation |
Random Generalized Inverse Wishart Distributed Matrices
Description
Generate n random matrices, distributed according
to the generalized inverse 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. Then the generalized inverse
Wishart distribution is the natural extension of the inverse Wishart using
the Moore-Penrose pseudo-inverse. This can generate samples for positive
semi-definite \Sigma
however, a function dedicated to generating
singular normal random distributions or singular pseudo Wishart distributions
should be used if that is desired.
Note there are different ways of parameterizing the Inverse
Wishart distribution, so check which one you need.
Here, if X \sim IW_p(\Sigma, \nu)
then
X^{-1} \sim W_p(\Sigma^{-1}, \nu)
.
Dawid (1981) has a different definition: if
X \sim W_p(\Sigma^{-1}, \nu)
and
\nu > p - 1
, then
X^{-1} = Y \sim IW(\Sigma, \delta)
,
where \delta = \nu - p + 1
.
Usage
rGenInvWishart(n, df, Sigma)
Arguments
n |
integer sample size. |
df |
integer parameter, "degrees of freedom", should be less than the
dimension of |
Sigma |
positive semi-definite |
Value
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)
.
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.
Bodnar, T., Mazur, S., Podgórski, K. "Singular inverse Wishart distribution and its application to portfolio theory", Journal of Multivariate Analysis, Volume 143, 2016, Pages 314-326, ISSN 0047-259X, doi: 10.1016/j.jmva.2015.09.021.
Bodnar, T., Okhrin, Y., "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions", Journal of Multivariate Analysis, Volume 99, Issue 10, 2008, Pages 2389-2405, ISSN 0047-259X, doi: 10.1016/j.jmva.2008.02.024.
Uhlig, Harald. "On Singular Wishart and Singular Multivariate Beta Distributions." Ann. Statist. 22 (1994), no. 1, 395–405. doi: 10.1214/aos/1176325375.
See Also
rWishart
, rInvWishart
,
and rPseudoWishart
Examples
set.seed(20181228)
A <- rGenInvWishart(1L, 4L, 5.0 * diag(5L))[, , 1]
A
# A should be singular
eigen(A)$values
set.seed(20181228)
B <- rPseudoWishart(1L, 4L, 5.0 * diag(5L))[, , 1]
# A should be a Moore-Penrose pseudo-inverse of B
B
# this should be equal to B
B %*% A %*% B
# this should be equal to A
A %*% B %*% A