wishmom (version 1.1.0)

iwishmom: Expectation of a Matrix-valued Function of an Inverse 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

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

Value

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

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)

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