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 \(\delta = \nu - p + 1\).
rGenInvWishart(n, df, Sigma)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 .
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 .
set.seed(20181228)
A <- rGenInvWishart(1L, 4L, 5.0 * diag(5L))[, , 1]
A
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.03316896 -0.084864096 0.003831550 0.031486573 0.060802330
#> [2,] -0.08486410 0.265497148 0.003410527 -0.059750288 -0.159210057
#> [3,] 0.00383155 0.003410527 0.025737818 -0.002461811 0.005335016
#> [4,] 0.03148657 -0.059750288 -0.002461811 0.045360861 0.054727822
#> [5,] 0.06080233 -0.159210057 0.005335016 0.054727822 0.139802644
# A should be singular
eigen(A)$values
#> [1] 4.227578e-01 4.247200e-02 2.813051e-02 1.620710e-02 8.448744e-17
set.seed(20181228)
B <- rPseudoWishart(1L, 4L, 5.0 * diag(5L))[, , 1]
# A should be a Moore-Penrose pseudo-inverse of B
B
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 6.855443 -1.979593 9.123138 12.366797 -10.283094
#> [2,] -1.979593 11.310039 -2.313472 1.767831 13.168709
#> [3,] 9.123138 -2.313472 38.453046 1.981594 -8.875384
#> [4,] 12.366797 1.767831 1.981594 33.137945 -16.467918
#> [5,] -10.283094 13.168709 -8.875384 -16.467918 33.403804
# this should be equal to B
B %*% A %*% B
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 6.855443 -1.979593 9.123138 12.366797 -10.283094
#> [2,] -1.979593 11.310039 -2.313472 1.767831 13.168709
#> [3,] 9.123138 -2.313472 38.453046 1.981594 -8.875384
#> [4,] 12.366797 1.767831 1.981594 33.137945 -16.467918
#> [5,] -10.283094 13.168709 -8.875384 -16.467918 33.403804
# this should be equal to A
A %*% B %*% A
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.03316896 -0.084864096 0.003831550 0.031486573 0.060802330
#> [2,] -0.08486410 0.265497148 0.003410527 -0.059750288 -0.159210057
#> [3,] 0.00383155 0.003410527 0.025737818 -0.002461811 0.005335016
#> [4,] 0.03148657 -0.059750288 -0.002461811 0.045360861 0.054727822
#> [5,] 0.06080233 -0.159210057 0.005335016 0.054727822 0.139802644