Description Usage Arguments Details Value Warning Note Examples
Samples a matrix Beta type II distribution.
1 2  rmatrixbetaII(n, p, a, b, Theta1 = NULL, Theta2 = NULL, def = 1,
checkSymmetry = TRUE)

n 
sample size, a positive integer 
p 
dimension, a positive integer 
a, b 
parameters of the distribution, positive numbers with constraints given in Details 
Theta1 
numerator noncentrality parameter, a positive semidefinite real
matrix of order 
Theta2 
denominator noncentrality parameter, a positive semidefinite real
matrix of order 
def 

checkSymmetry 
logical, whether to check the symmetry of 
A Beta type II random matrix V is defined as follows. Take two independent Wishart random matrices S_{1} ~ W_{p}(2a,I_{p},Θ_{1}) and S_{2} ~ W_{p}(2b,I_{p},Θ_{2}).
definition 1: V = S_{2}^{½}S_{1}S_{2}^{½}
definition 2: V = S_{1}^{½}S_{2}^{1}S_{1}^{½}
In the central case, the two definitions yield the same distribution. Under definition 2, the Beta type II distribution is related to the Beta distribution by V ~ U(IU)^{1}.
Parameters a
and b
are positive numbers that satisfy the
following constraints:
in any case, b > (p1)/2
if Theta1
is the null matrix and a < (p1)/2
, then
a
must be half an integer
if Theta1
is not the null matrix, a >= (p1)/2
A numeric threedimensional array; simulations are stacked along the third dimension (see example).
The issue described in the Warning section of rmatrixbeta
also concerns rmatrixbetaII
.
The matrix variate Beta distribution of type II is usually defined only for a > (p1)/2 and b > (p1)/2. In this case, a random matrix V generated from this distribution satisfies V > 0. For an half integer a ≤ (p1)/2, it satisfies V ≥ 0 and rank(V)=2a.
1 2  Bsims < rmatrixbetaII(10000, 3, 1, 1.5)
dim(Bsims) # 3 3 10000

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