wishmom: Expectation of a Matrix-valued Function of a beta-Wishart...

In wishmom: Compute Moments Related to Beta-Wishart and Inverse Beta-Wishart Distributions

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

Description

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}]

Usage

wishmom(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 (0 by default)
alpha	The type of Wishart distribution (\alpha=2/\beta): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

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)^4] 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)
wishmom(n, S, 4) # iw = 0, for real Wishart distribution

# Example 2: For E[tr(W)^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)
wishmom(n, S, c(2, 0, 1), 2, 2) # iw = 2, for real Wishart distribution

# Example 3: For E[tr(W)^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)
wishmom(n, S, c(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution

# Example 4: For E[tr(W)*tr(W^2)^2*tr(W^3)^2*W] with W ~ W_m^2(n,S),
# where n and S are defined below:
n <- 20
S <- matrix(c(25, 49 + 2i,
              49 - 2i, 109), nrow=2, ncol=2)
wishmom(n, S, c(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution

wishmom documentation built on Sept. 11, 2024, 8:29 p.m.