rmatrixbetaII {matrixsampling}

Matrix Beta II sampler

Description

Samples a matrix Beta type II distribution.

Usage

rmatrixbetaII(n, p, a, b, Theta1 = NULL, Theta2 = NULL, def = 1,
  checkSymmetry = TRUE)

Arguments

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 p; setting it to NULL (default) is equivalent to setting it to the zero matrix

Theta2

denominator noncentrality parameter, a positive semidefinite real matrix of order p; setting it to NULL (default) is equivalent to setting it to the zero matrix

def

1 or 2, the definition used; see Details

checkSymmetry

logical, whether to check the symmetry of Theta1 and Theta2

Details

A Beta type II random matrix V is defined as follows. Take two independent Wishart random matrices S₁ ~ W_p(2a,I_p,Θ₁) and S₂ ~ W_p(2b,I_p,Θ₂).

• definition 1: V = S₂^-½S₁S₂^-½

• definition 2: V = S₁^½S₂^-1S₁^½

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(I-U)^-1.

Parameters a and b are positive numbers that satisfy the following constraints:

• in any case, b > (p-1)/2

• if Theta1 is the null matrix and a < (p-1)/2, then a must be half an integer

• if Theta1 is not the null matrix, a >= (p-1)/2

Value

A numeric three-dimensional array; simulations are stacked along the third dimension (see example).

Warning

The issue described in the Warning section of rmatrixbeta also concerns rmatrixbetaII.

Note

The matrix variate Beta distribution of type II is usually defined only for a > (p-1)/2 and b > (p-1)/2. In this case, a random matrix V generated from this distribution satisfies V > 0. For an half integer a \le (p-1)/2, it satisfies V \ge 0 and rank(V)=2a.

Examples

Bsims <- rmatrixbetaII(10000, 3, 1, 1.5)
dim(Bsims) # 3 3 10000