iwishmom function

Expectation of a Matrix-valued Function of an Inverse beta-Wishart Distribution

When iw = 0, the function calculates $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$, where $W \sim W_m^{\beta}(n, S)$. When iw != 0, the function calculates $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.

iwishmom(n, S, f, iw = 0, alpha = 2)

Arguments

• n: The degrees of freedom of the beta-Wishart matrix $W$

• S: The covariance matrix of the beta-Wishart matrix $W$

• f: A vector of nonnegative integers $f_j$ that represents the power of $\mbox{tr}(W^{-j})$, where $j=1, \ldots, r$

• iw: The power of the inverse beta-Wishart matrix $W^{-1}$ (0 by default)

• alpha: The type of Wishart distribution $(\alpha=2/\beta)$:

  • 1/2: Quaternion Wishart
  • 1: Complex Wishart
  • 2: Real Wishart (default)

Returns

When iw = 0, it returns $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$. When iw != 0, it returns $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.

Examples

# Example 1: For E[tr(W^{-1})^2] with W ~ W_m^1(n,S),
# where n and S are defined below:
n <- 20
S <- matrix(c(25, 49,
              49, 109), nrow=2, ncol=2)
iwishmom(n, S, 2) # iw = 0, for real Wishart distribution

# Example 2: For E[tr(W^{-1})^2*tr(W^{-3})W^{-2}] with W ~ W_m^1(n,S),
# where n and S are defined below:
n <- 20
S <- matrix(c(25, 49,
              49, 109), nrow=2, ncol=2)
iwishmom(n, S, c(2, 0, 1), 2, 2) # iw = 2, for real Wishart distribution

# Example 3: For E[tr(W^{-1})^2*tr(W^{-3})] with W ~ W_m^2(n,S),
# where n and S are defined below:
# Hermitian S for the complex case
n <- 20
S <- matrix(c(25, 49 + 2i,
              49 - 2i, 109), nrow=2, ncol=2)
iwishmom(n, S, c(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution

# Example 4: For E[tr(W^{-1})*tr(W^{-2})^2*tr(W^{-3})^2*W^{-1}] with W ~ W_m^2(n,S),
# where n and S are defined below:
n <- 30
S <- matrix(c(25, 49 + 2i,
              49 - 2i, 109), nrow=2, ncol=2)
iwishmom(n, S, c(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution