wishmom_sym: Symbolic Expectation of a Matrix-valued Function of a...

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

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

Description

When iw = 0, the function returns an analytical expression of E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}], where W \sim W_m^{\beta}(n, S). When iw != 0, the function returns an analytical expression of E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]. For a given f, iw, and alpha, this function provides the aforementioned expectations in terms of the variables n and \Sigma.

Usage

wishmom_sym(f, iw = 0, alpha = 2, latex = FALSE)

Arguments

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 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)
latex	A Boolean indicating whether the output will be a LaTeX string or a dataframe (FALSE by default)

Value

When iw = 0, it returns an analytical expression of E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]. When iw != 0, it returns an analytical expression of E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]. If latex = FALSE, the output is a data frame that stores the coefficients for calculating the result. If latex = TRUE, the output is a LaTeX formatted string of the result in terms of n and \Sigma.

Examples

# Example 1: For E[tr(W)^4] with W ~ W_m^1(n,Sigma), represented as a dataframe:
wishmom_sym(4) # iw = 0, for real Wishart distribution

# Example 2: For E[tr(W)*tr(W^2)W] with W ~ W_m^1(n,S), represented as a dataframe:
wishmom_sym(c(1, 1), 1) # iw = 1, for real Wishart distribution

# Example 3: For E[tr(W)^4] with W ~ W_m^2(n,S), represented as a LaTeX string:
# Using writeLines() to format
writeLines(wishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution

# Example 4: For E[tr(W)*tr(W^2)W] with W ~ W_m^2(n,S), represented as a LaTeX string:
# Using writeLines() to format
writeLines(wishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for real Wishart distribution

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