# rmatrixbetaII: Matrix Beta II sampler In matrixsampling: Simulations of Matrix Variate Distributions

## Description

Samples a matrix Beta type II distribution.

## Usage

 ```1 2``` ```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 ~ Wp(2a,Ip1) and S2 ~ Wp(2b,Ip2).

• definition 1: V = S2S1S2

• definition 2: V = S1½S2-1S1½

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 ≤ (p-1)/2, it satisfies V ≥ 0 and rank(V)=2a.

## Examples

 ```1 2``` ```Bsims <- rmatrixbetaII(10000, 3, 1, 1.5) dim(Bsims) # 3 3 10000 ```

matrixsampling documentation built on Aug. 25, 2019, 1:03 a.m.