iwishmom_sym function

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

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 $\tilde{n}$ and $\Sigma$.

iwishmom_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 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)

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

Returns

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 $\tilde{n}$ and $\Sigma$.

Examples

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

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

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

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