Mathematical Background

$β$ -Wishart Distribution

The $β$ -Wishart distribution is a fundamental distribution in multivariate statistics. When $β = 1, 2, 4, 8$ , it is the real Wishart, complex Wishart, quaternion Wishart, and octonion Wishart, respectively. The density function of $W \sim W_{m}^{β} (n, Σ)$ , i,e., a $β$ -Wishart distribution with $n$ degrees of freedom and an $m \times m$ covariance matrix $Σ$ , is given by (when $n > m - 1$ ) (see Díaz-García and Gutiérrez-Jáimez (2011, Corollary 1))

$f (W) = \frac{{(\frac{β}{2})}^{\frac{m n β}{2}}}{Γ_{m}^{(β)} (\frac{n β}{2}) | Σ |^{\frac{n β}{2}}} | W |^{\frac{(n - m + 1) β}{2} - 1} etr (- \frac{β Σ^{- 1} W}{2}),$

where

$Γ_{m}^{(β)} (a) = π^{\frac{m (m - 1) β}{4}} \prod_{i = 1}^{m} Γ (a - \frac{(i - 1) β}{2}) .$

Note that we do not require $n$ to be an integer but the definition of the density of $W$ requires $β = 1, 2, 4$ or $8$ . However, if our interest is only on the functions of eigenvalues of $W$ , we can generalize this to any real $β > 0$ . Therefore, for any symmetric functions (say power-sum) of the eigenvalues of $W$ , they can be well defined even when $β$ is not equal to $1, 2, 4$ or $8$ . For $W \sim W_{m}^{β} (n, Σ)$ , the joint density of its eigenvalues $λ_{1} \geq \dots \geq λ_{m}$ is given by (see Dresnky, Edelman, Genoar, Kan, and Koev (2021))

$f (λ_{1}, \dots, λ_{m}) = \frac{| Σ |^{- \frac{n β}{2}}}{K_{m}^{(β)} (\frac{n β}{2})} | Λ |^{\frac{(n - m + 1) β}{2} - 1}_{0}^{} F_{0}^{(β)} (- \frac{β}{2} Λ, Σ^{- 1}) \prod_{1 \leq i < j \leq m} (λ_{i} - λ_{j})^{β},$ where $Λ = Diag (λ_{1}, \dots, λ_{m})$ ,

$K_{m}^{(β)} (a) = \frac{{(\frac{2}{β})}^{m a}}{π^{\frac{m (m - 1) β}{2}}} \frac{Γ_{m}^{(β)} (\frac{m β}{2}) Γ_{m}^{(β)} (a)}{{[Γ (\frac{β}{2})]}^{m}},$

$_{0}^{} F_{0}^{(β)} (A, B) = \sum_{k = 0}^{\infty} \sum_{κ ⊢ k} \frac{C_{κ}^{(β)} (A) C_{κ}^{(β)} (B)}{k! C_{κ}^{(β)} (I_{m})},$

and $C_{κ}^{(β)} (X)$ is the Jack function of the eigenvalues of $X$ .

Instead of using $β$ , we use $α = 2 / β$ in our programs to describe the type of Wishart distribution. Therefore, $α = 2$ is for real Wishart, $α = 1$ is for complex Wishart, and $α = 1 / 2$ is for quaternion Wishart.

Moments of Matrix-valued Functions of $β$ -Wishart and Inverse $β$ -Wishart Distributions

Let $λ = (λ_{1}, \dots, λ_{k})$ be an integer partition of a positive integer $k$ , where $| λ | = λ_{1} + \dots + λ_{k} = k$ , with $λ_{1} \geq λ_{2} \geq \dots \geq λ_{k} \geq 0$ . The power-sum symmetric function $p_{λ} (W)$ of $W$ corresponding to a partition $λ$ is defined as

$p_{λ} (W) = \prod_{i = 1}^{ℓ (λ)} p_{λ_{i}} (W),$

where $ℓ (λ)$ is the number of non-zero parts of $λ$ , and $p_{i} (W) = tr (W^{i})$ . We are interested in computing

$E [W^{r} p_{λ} (W)] and E [W^{- r} p_{λ} (W^{- 1})],$

where $W \sim W_{m}^{β} (n, Σ)$ .

The method that we use is based on a generalization of the recurrence relations given in Hillier and Kan (2024) for which the cases of $β = 1$ and $2$ were developed. Specifically, we have the following recurrence relations:

$\begin{aligned} E [W^{r + 1} p_{λ} (W)] & = [n + (\frac{2}{β} - 1) r] Σ E [W^{r} p_{λ} (W)] + \sum_{j = 1}^{r} Σ E [W^{r - j} p_{j} (W) p_{λ} (W)] \\ + \frac{2}{β} \sum_{i = 1}^{ℓ (λ)} λ_{i} Σ E [W^{r + λ_{i}} p_{λ_{(i)}} (W)], \\ Σ^{- 1} E [W^{- r} p_{λ} (W^{- 1})] & = [\tilde{n} - (\frac{2}{β} - 1) r] E [W^{- (r + 1)} p_{λ} (W^{- 1})] - \sum_{j = 1}^{r} E [W^{- r - 1 + j} p_{j} (W^{- 1}) p_{λ} (W^{- 1})] \\ - \frac{2}{β} \sum_{i = 1}^{ℓ (λ)} λ_{i} E [W^{- r - 1 - λ_{i}} p_{λ_{(i)}} (W^{- 1})], \end{aligned}$ where $\tilde{n} = n - m + 1 - (2 / β)$ and $λ_{(i)}$ is $λ$ with its $i$ -th element removed. Together with the boundary conditions $E [W] = n Σ$ and $E [W^{- 1}] = Σ^{- 1} / \tilde{n}$ , we can obtain $E [E^{r} p_{λ} (W)]$ and $E [W^{- r} p_{λ} (W^{- 1})]$ . Note that $E [W^{- r} p_{λ} (W^{- 1})]$ exists if and only if $\tilde{n} > 2 (r + | λ |)$ .

Let $k = r + | λ |$ . Hillier and Kan (2024) show that $\begin{aligned} E [W^{r} p_{λ} (W)] & = \sum_{i = 1}^{k} [\sum_{ρ ⊢ k - i} c_{λ, ρ} p_{ρ} (Σ)] Σ^{i}, \\ E [W^{- r} p_{λ} (W^{- 1})] & = \sum_{i = 1}^{k} [\sum_{ρ ⊢ k - i} {\tilde{c}}_{λ, ρ} p_{ρ} (Σ^{- 1})] Σ^{- i}, \end{aligned}$ where $c_{λ, ρ}$ and ${\tilde{c}}_{λ, ρ}$ are constants that depend on $n$ and $\tilde{n}$ , respectively, but they do not depend on $Σ$ . In addition, we have $\begin{aligned} E [p_{λ} (W)] & = \sum_{κ ⊢ k} h_{κ} p_{κ} (Σ), \\ E [p_{λ} (W^{- 1})] & = \sum_{κ ⊢ k} {\tilde{h}}_{κ} p_{κ} (Σ^{- 1}), \end{aligned}$ where $h_{κ}$ and ${\tilde{h}}_{κ}$ are constants that depend on $n$ and $\tilde{n}$ , repsectively, but they do not depend on $Σ$ .

Using the recurrence relations, Hillier and Kan (2024) develop efficient algorithms for computing the constants $c_{λ, ρ}$ , ${\tilde{c}}_{λ, ρ}$ , $h_{κ}$ and ${\tilde{h}}_{κ}$ .

Main Functions in the Package

There are four main functions in this package: wishmom(), iwishmom(), wishmom_sym(), and iwishmom_sym(). The first two are used to numerically compute $E [W^{r} p_{λ} (W)]$ and $E [W^{- r} p_{λ} (W^{- 1})]$ , respectively. The last two are used to generate an analytical expression of $E [W^{r} p_{λ} (W)]$ and $E [W^{- r} p_{λ} (W^{- 1})]$ , respectively.

Moments of $β$ -Wishart:

wishmom()

The function wishmom() computes $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}} W^{i w}]$ numerically, where $W \sim W_{m}^{β} (n, Σ)$ . When $i w = 0$ , it computes $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}}]$ .

Arguments

n: degrees of freedom of the $β$ -Wishart distribution
S: covariance matrix of the $β$ -Wishart distribution
f: a vector of nonnegative integers $f_{j}$ that represents the power for $tr (W^{j})$ , $j = 1, \dots, r$
iw: Power of $W$
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

When $i w = 0$ , it returns $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}}]$ . When $i w \neq 0$ , it returns $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}} W^{i w}]$ .

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
#> [1] 8.705084e+13

# 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
#>              [,1]         [,2]
#> [1,] 9.039462e+23 1.956948e+24
#> [2,] 1.956948e+24 4.258714e+24

# Example 3: For E[tr(W)^2*tr(W^3)] 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(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution
#> [1] 2.078126e+17

# Example 4: For E[tr(W)*tr(W^2)^2*tr(W^3)^2*W] with W ~ W_m^2(n,S), where n, 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
#>                            [,1]                       [,2]
#> [1,] 3.418999e+41+5.014362e+20i 6.943130e+41-2.833930e+40i
#> [2,] 6.943130e+41+2.833930e+40i 1.532151e+42-2.882805e+22i

wishmom_sym()

The function wishmom_sym() generates an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}} W^{i w}]$ , where $W \sim W_{m}^{β} (n, Σ)$ . When $i w = 0$ , it generates an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}}]$ .

Arguments

f: a vector of nonnegative integers $f_{j}$ that represents the power for $tr (W^{j})$ , $j = 1, \dots, r$
iw: Power of $W$
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)
latex: The type of output
- TRUE: LaTeX expression
- FALSE: Dataframe (default)

Output

When $i w = 0$ , the output is an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}}]$ . When $i w \neq 0$ , the output is an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{j})^{f_{j}} W^{i w}]$ . If latex = FALSE (default), the output is a data frame that stores the coefficients for the analytical expression. If latex = TRUE, the output is a $L A T E X$ formatted string of the result in terms of $n$ and $Σ$ .

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
#>     kappa h_kappa
#> 1       4      48
#> 2     3,1     32n
#> 3     2,2     12n
#> 4   2,1,1   12n^2
#> 5 1,1,1,1     n^3

# Example 2: For E[tr(W)*tr(W^2)W] with W ~ W_m^1(n,Sigma), represented as a dataframe:
wishmom_sym(c(1,1), 1) # iw = 1, for real Wishart distribution
#>   i   rho          c
#> 1 1     3    4n^2+4n
#> 2 1   2,1 n^3+n^2+4n
#> 3 1 1,1,1        n^2
#> 4 2     2  2n^2+2n+8
#> 5 2   1,1         6n
#> 6 3     1 4n^2+4n+16
#> 7 4  <NA>     24n+24

# Example 3: For E[tr(W)^4] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(wishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution
#> (6)p_{(4)}(\Sigma)
#> +(8n)p_{(3,1)}(\Sigma)
#> +(3n)p_{(2,2)}(\Sigma)
#> +(6n^2)p_{(2,1,1)}(\Sigma)
#> +(n^3)p_{(1,1,1,1)}(\Sigma)

# Example 4: For E[tr(W)*tr(W^2)W] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(wishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for complex Wishart distribution
#> [(2n^2)p_{(3)}(\Sigma)+(n^3+2n)p_{(2, 1)}(\Sigma)+(n^2)p_{(1, 1, 1)}(\Sigma)]\Sigma
#> +[(n^2+2)p_{(2)}(\Sigma)+(3n)p_{(1, 1)}(\Sigma)]\Sigma^2
#> +(2n^2+4)p_{(1)}(\Sigma)\Sigma^3
#> +(6n)\Sigma^4

Moments of Inverse $β$ -Wishart:

iwishmom()

The function iwishmom() computes $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}} W^{- i w}]$ numerically, where $W \sim W_{m}^{β} (n, Σ)$ . When $i w = 0$ , it computes $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}}]$ .

Arguments

n: degrees of freedom of the $β$ -Wishart distribution
S: covariance matrix of the $β$ -Wishart distribution
f: a vector of nonnegative integers $f_{j}$ that represents the power for $tr (W^{- j})$ , $j = 1, \dots, r$
iw: Power of $W^{- 1}$
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

When $i w = 0$ , it returns $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}}]$ . When $i w \neq 0$ , it returns $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}} W^{- i w}]$ .

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
#> [1] 0.0006680892

# 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
#>               [,1]          [,2]
#> [1,]  1.328434e-10 -6.101692e-11
#> [2,] -6.101692e-11  2.824292e-11

# 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:
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
#> [1] 1.17985e-08

# 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
#>                             [,1]                        [,2]
#> [1,]  1.348928e-21+0.000000e+00i -6.116211e-22+2.496413e-23i
#> [2,] -6.116211e-22-2.496413e-23i  3.004350e-22+0.000000e+00i

iwishmom_sym()

The function iwishmom_sym() generates an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}} W^{- i w}]$ , where $W \sim W_{m}^{β} (n, Σ)$ . When $i w = 0$ , it generates an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}}]$ .

Arguments

f: a vector of nonnegative integers $f_{j}$ that represents the power for $tr (W^{- j})$ , $j = 1, \dots, r$
iw: Power of $W^{- 1}$
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)
latex: The type of output
- TRUE: LaTeX expression
- FALSE: Dataframe (default)

Output

When $i w = 0$ , the output is an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}}]$ . When $i w \neq 0$ , the output is an analytical expression of $E [\prod_{j = 1}^{r} tr (W^{- j})^{f_{j}} W^{- i w}]$ . If latex = FALSE (default), the output is a data frame that stores the coefficients for the analytical expression. If latex = TRUE, the output is a $L A T E X$ formatted string of the result in terms of $\tilde{n}$ and $Σ$ , where $\tilde{n} = n - m + 1 - α$ and $m$ is the dimension of the $β$ -Wishart distribution.

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
#> $dataframe
#>     kappa      h_kappa_numerator
#> 1       4              240n1-288
#> 2     3,1           64n1^2-256n1
#> 3     2,2        12n1^2-60n1+216
#> 4   2,1,1  12n1^3-72n1^2+36n1+72
#> 5 1,1,1,1 n1^4-7n1^3+n1^2+35n1-6
#> 
#> $denominator
#> [1] "n1^8-7n1^7-11n1^6+107n1^5+34n1^4-388n1^3-24n1^2+288n1"

# Example 2: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^1(n,Sigma), represented as a dataframe:
iwishmom_sym(c(1,1), 1) # iw = 1, for real Wishart distribution
#> $dataframe
#>   i   rho          c_numerator
#> 1 1     3 4n1^3-16n1^2-20n1+24
#> 2 1   2,1 n1^4-6n1^3+3n1^2+6n1
#> 3 1 1,1,1     n1^3-6n1^2+3n1+6
#> 4 2     2    2n1^3-10n1^2+36n1
#> 5 2   1,1       10n1^2-42n1+36
#> 6 3     1 4n1^3-16n1^2+60n1-72
#> 7 4  <NA>          40n1^2-48n1
#> 
#> $denominator
#> [1] "n1^8-7n1^7-11n1^6+107n1^5+34n1^4-388n1^3-24n1^2+288n1"

# Example 3: For E[tr(W^{-1})^4] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(iwishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution
#> [(30\tilde{n})p_{(4)}(\Sigma^{-1})
#> +(16\tilde{n}^2-24)p_{(3,1)}(\Sigma^{-1})
#> +(3\tilde{n}^2+18)p_{(2,2)}(\Sigma^{-1})
#> +(6\tilde{n}^3-24\tilde{n})p_{(2,1,1)}(\Sigma^{-1})
#> +(\tilde{n}^4-8\tilde{n}^2+6)p_{(1,1,1,1)}(\Sigma^{-1})]
#> /(\tilde{n}^8-14\tilde{n}^6+49\tilde{n}^4-36\tilde{n}^2)

# Example 4: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(iwishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for complex Wishart distribution
#> [[(2\tilde{n}^3-8\tilde{n})p_{(3)}(\Sigma^{-1})+(\tilde{n}^4-4\tilde{n}^2)p_{(2,1)}(\Sigma^{-1})+(\tilde{n}^3-4\tilde{n})p_{(1,1,1)}(\Sigma^{-1})]\Sigma^{-1}
#> +[(\tilde{n}^3+6\tilde{n})p_{(2)}(\Sigma^{-1})+(5\tilde{n}^2)p_{(1,1)}(\Sigma^{-1})]\Sigma^{-2}
#> +(2\tilde{n}^3+12\tilde{n})p_{(1)}(\Sigma^{-1})\Sigma^{-3}
#> +(10\tilde{n}^2)\Sigma^{-4}]
#> /(\tilde{n}^8-14\tilde{n}^6+49\tilde{n}^4-36\tilde{n}^2)

Auxiliary Functions

Below is a list of auxiliary functions that are called by wishmom, iwishmom, wishmom_sym, and iwishmom_sym.

ip_desc()

The function ip_desc() generates all integer partitions of a given integer k in a reverse lexicographical order.

Arguments

k: A positive integer to be partitioned

Output

A matrix where each row represents an integer partition of k, listed in a reverse lexicographical order.

Examples

# Example 1: List of integer partitions of 3
ip_desc(3)
#>      [,1] [,2] [,3]
#> [1,]    3    0    0
#> [2,]    2    1    0
#> [3,]    1    1    1

# Example 2: List of integer partitions of 5
ip_desc(5)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    5    0    0    0    0
#> [2,]    4    1    0    0    0
#> [3,]    3    2    0    0    0
#> [4,]    3    1    1    0    0
#> [5,]    2    2    1    0    0
#> [6,]    2    1    1    1    0
#> [7,]    1    1    1    1    1

dkmap()

The function dkmap() computes the mapping matrix $D_{k}$ discussed in Appendix B of Hillier and Kan (2024), modified for the general $β$ -Wishart case. The returned matrix is $D_{k}$ but with $n$ in the diagonal elements removed.

Arguments

k: The order of the mapping matrix $D_{k}$ (a positive integer)
alpha: The type of $β$ -Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

The mapping matrix $D_{k}$ but with $n$ removed from its diagonal.

Examples

# Example 1: Compute the mapping matrix for k = 2, real Wishart
dkmap(2)
#>      [,1] [,2] [,3] [,4]
#> [1,]    2    1    1    0
#> [2,]    2    1    0    1
#> [3,]    4    0    0    0
#> [4,]    0    4    0    0

# Example 2: Compute the mapping matrix for k = 1, complex Wishart
dkmap(1, 1)
#>      [,1] [,2]
#> [1,]    0    1
#> [2,]    1    0

# Example 3: Compute the mapping matrix for k = 2, quaternion Wishart
dkmap(2, 1/2)
#>      [,1] [,2] [,3] [,4]
#> [1,] -1.0  1.0    1    0
#> [2,]  0.5 -0.5    0    1
#> [3,]  1.0  0.0    0    0
#> [4,]  0.0  1.0    0    0

denpoly()

The function denpoly() computes the coefficients of the denominator polynomial for the elements ${\tilde{H}}_{k}$ and ${\tilde{C}}_{k}$ . The function returns a vector containing the coefficients in descending powers of $\tilde{n}$ , with the last element being the coefficient of $\tilde{n}$ .

Arguments

k: The order of the polynomial (a positive integer)
alpha: The type of $β$ -Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

A vector containing the coefficients of the denominator polynomial in descending powers of $\tilde{n}$ for the elements of ${\tilde{H}}_{k}$ and ${\tilde{C}}_{k}$ .

Examples

# Example 1: Compute the denominator polynomial for k = 3 and alpha = 2
# Output corresponds to the polynomial n1^5-3n1^4-8n1^3+12n1^2+16n1,
# where n1 is \eqn{\tilde{n}}
denpoly(3)
#> [1]  1 -3 -8 12 16

# Example 2: Compute the denominator polynomial for k = 2 and alpha = 1
# Output corresponds to the polynomial n1^3-n1, where n1 is \eqn{\tilde{n}}
denpoly(2, alpha = 1)
#> [1]  1  0 -1

qk_coeff()

The function qk_coeff() computes the coefficient matrix $C_{k}$ , which is obtained based on Corollary 1 of Hillier and Kan (2024), after a modification for the general $β$ -Wishart case. $C_{k}$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$ .

Arguments

k: The order of the $C_{k}$ matrix
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

A 3-dimensional array representing $C_{k}$ , a matrix of constants that allow us to obtain $E [p_{λ} (W) W^{r}]$ , where $r + | λ | = k$ and $W \sim W_{m}^{β} (n, Σ)$ .

Examples

# Example 1:
qk_coeff(2) # For real Wishart distribution with k = 2
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]    1    1
#> [2,]    2    0

# Example 2:
qk_coeff(3, 1) # For complex Wishart distribution with k = 3
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    2    1    0
#> [2,]    2    0    0    1
#> [3,]    2    0    0    1
#> [4,]    0    2    1    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    1
#> [2,]    0    1    1    0
#> [3,]    0    2    0    0
#> [4,]    2    0    0    0

# Example 3:
qk_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,] -0.5    1
#> [2,]  0.5    0

wish_ps()

The function wish_ps() computes the coefficient matrix $H_{k}$ that allows us to compute $E [p_{κ} (W)]$ , which is obtained based on Proposition 5 of Hillier and Kan (2024), after a modification for the general $β$ -Wishart case. $H_{k}$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$ .

Arguments

k: The order of the $H_{k}$ matrix
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

A 3-dimensional array representing $H_{k}$ , a matrix of constants that allows us to obtain $E [p_{κ} (W)]$ , where $| κ | = k$ and $W \sim W_{m}^{β} (n, Σ)$ .

Examples

# Example 1:
wish_ps(3) # For real Wishart distribution with k = 3
#> , , 1
#> 
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3]
#> [1,]    3    3    0
#> [2,]    4    1    1
#> [3,]    0    6    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3]
#> [1,]    4    3    1
#> [2,]    4    4    0
#> [3,]    8    0    0

# Example 2:
wish_ps(4, 1) # For complex Wishart distribution with k = 4
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0    0    0
#> [2,]    0    1    0    0    0
#> [3,]    0    0    1    0    0
#> [4,]    0    0    0    1    0
#> [5,]    0    0    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    4    2    0    0
#> [2,]    3    0    0    3    0
#> [3,]    4    0    0    2    0
#> [4,]    0    4    1    0    1
#> [5,]    0    0    0    6    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    5    0    0    6    0
#> [2,]    0    7    3    0    1
#> [3,]    0    8    2    0    1
#> [4,]    6    0    0    5    0
#> [5,]    0    8    3    0    0
#> 
#> , , 4
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    4    1    0    1
#> [2,]    3    0    0    3    0
#> [3,]    2    0    0    4    0
#> [4,]    0    4    2    0    0
#> [5,]    6    0    0    0    0

# Example 3:
wish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,] -0.5    1
#> [2,]  0.5    0

qkn_coeff()

The function qkn_coeff() computes the inverse of the coefficient matrix ${\tilde{C}}_{k}$ , which is obtained based on Corollary 2 of Hillier and Kan (2024), after a modification for the general $β$ -Wishart case. ${\tilde{C}}_{k}^{- 1}$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $\tilde{n}$ .

Arguments

k: The order of the ${\tilde{C}}_{k}$ matrix
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

A 3-dimensional array representing ${\tilde{C}}_{k}^{- 1}$ , a matrix of constants that allow us to obtain $E [p_{λ} (W^{- 1}) W^{- r}]$ , where $r + | λ | = k$ and $W \sim W_{m}^{β} (n, Σ)$ .

Examples

# Example 1:
qkn_coeff(2) # For real Wishart distribution with k = 2
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]   -1   -1
#> [2,]   -2    0

# Example 2:
qkn_coeff(3, 1) # For complex Wishart distribution with k = 3
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0   -2   -1    0
#> [2,]   -2    0    0   -1
#> [3,]   -2    0    0   -1
#> [4,]    0   -2   -1    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    1
#> [2,]    0    1    1    0
#> [3,]    0    2    0    0
#> [4,]    2    0    0    0

# Example 3:
qkn_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]  0.5   -1
#> [2,] -0.5    0

iwish_ps()

The function iwish_ps() computes the inverse of the coefficient matrix ${\tilde{H}}_{k}$ that allows us to compute $E [p_{κ} (W^{- 1})]$ , which is obtained based on Eq.(82) of Hillier and Kan (2024), after a modification for the general $β$ -Wishart case. ${\tilde{H}}_{k}^{- 1}$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $\tilde{n}$ .

Arguments

k: The order of the ${\tilde{H}}_{k}$ matrix
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

A 3-dimensional array representing ${\tilde{H}}_{k}^{- 1}$ , a matrix of constants that allows us to obtain $E [p_{κ} (W^{- 1})]$ , where $| κ | = k$ and $W \sim W_{m}^{β} (n, Σ)$ .

Examples

# Example 1:
iwish_ps(3) # For real Wishart distribution with k = 3
#> , , 1
#> 
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3]
#> [1,]   -3   -3    0
#> [2,]   -4   -1   -1
#> [3,]    0   -6    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3]
#> [1,]    4    3    1
#> [2,]    4    4    0
#> [3,]    8    0    0

# Example 2:
iwish_ps(4, 1) # For complex Wishart distribution with k = 4
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0    0    0
#> [2,]    0    1    0    0    0
#> [3,]    0    0    1    0    0
#> [4,]    0    0    0    1    0
#> [5,]    0    0    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0   -4   -2    0    0
#> [2,]   -3    0    0   -3    0
#> [3,]   -4    0    0   -2    0
#> [4,]    0   -4   -1    0   -1
#> [5,]    0    0    0   -6    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    5    0    0    6    0
#> [2,]    0    7    3    0    1
#> [3,]    0    8    2    0    1
#> [4,]    6    0    0    5    0
#> [5,]    0    8    3    0    0
#> 
#> , , 4
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0   -4   -1    0   -1
#> [2,]   -3    0    0   -3    0
#> [3,]   -2    0    0   -4    0
#> [4,]    0   -4   -2    0    0
#> [5,]   -6    0    0    0    0

# Example 3:
iwish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]  0.5   -1
#> [2,] -0.5    0

qkn_coeffr()

The function qkn_coeffr() computes the coefficient matrix ${\tilde{C}}_{k}$ for the general $β$ -Wishart case. Elements of ${\tilde{C}}_{k}$ are rational polynomials of $\tilde{n}$ . The output contains two components: c and den. c is a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the numerator polynomial in descending powers of $\tilde{n}$ , and den is a vector that represents the coefficients of the denominator polynomial in descending power of $\tilde{n}$ .

Arguments

k: The order of the ${\tilde{C}}_{k}$ matrix
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

The output has two components: c and den. c is a 3-dimensional array representing the numerator polynomial of ${\tilde{C}}_{k}$ , and den is a vector representing the denominator polynomial of ${\tilde{C}}_{k}$ , where ${\tilde{C}}_{k}$ is a matrix of constants that allow us to obtain $E [p_{λ} (W^{- 1}) W^{- r}]$ , where $r + | λ | = k$ and $W \sim W_{m}^{β} (n, Σ)$ .

Examples

# Example 1:
qkn_coeffr(2) # For real Wishart distribution with k = 2
#> $c
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]    0    1
#> [2,]    2   -1
#> 
#> 
#> $den
#> [1]  1 -1 -2
 
# Example 2:
qkn_coeffr(3, 1) # For complex Wishart distribution with k = 3
#> $c
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    2    1    0
#> [2,]    2    0    0    1
#> [3,]    2    0    0    1
#> [4,]    0    2    1    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    2
#> [2,]    0    0    2    0
#> [3,]    0    4   -2    0
#> [4,]    4    0    0   -2
#> 
#> 
#> $den
#> [1]  1  0 -5  0  4

# Example 3:
qkn_coeffr(2, 1/2) # For quaternion Wishart distribution with k = 2
#> $c
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]  0.0  1.0
#> [2,]  0.5  0.5
#> 
#> 
#> $den
#> [1]  1.0  0.5 -0.5

iwish_psr()

The function iwish_psr() computes the coefficient matrix ${\tilde{H}}_{k}$ for the general $β$ -Wishart case. Elements of ${\tilde{H}}_{k}$ are rational polynomials of $\tilde{n}$ . The output contains two components: c and den. c is a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the numerator polynomial in descending powers of $\tilde{n}$ , and den is a vector that represents the coefficients of the denominator polynomial in descending power of $\tilde{n}$ .

Arguments

k: The order of the ${\tilde{C}}_{k}$ matrix
alpha: The type of Wishart distribution ( $α = 2 / β$ ):
- 1/2: Quaternion Wishart
- 1: Complex Wishart
- 2: Real Wishart (default)

Output

The output has two components: c and den. c is a 3-dimensional array representing the numerator polynomial of ${\tilde{H}}_{k}$ , and den is a vector representing the denominator polynomial of ${\tilde{H}}_{k}$ , where ${\tilde{H}}_{k}$ is a matrix of constants that allow us to obtain $E [p_{λ} (W^{- 1})]$ , where $| λ | = k$ and $W \sim W_{m}^{β} (n, Σ)$ .

Examples

# Example 1:
iwsih_psr(2) # For real Wishart distribution with k = 2
#> $c
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]    0    1
#> [2,]    2   -1
#> 
#> 
#> $den
#> [1]  1 -1 -2
 
# Example 2:
iwish_psr(4, 1) # For complex Wishart distribution with k = 4
#> $c
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0    0    0
#> [2,]    0    1    0    0    0
#> [3,]    0    0    1    0    0
#> [4,]    0    0    0    1    0
#> [5,]    0    0    0    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    4    2    0    0
#> [2,]    3    0    0    3    0
#> [3,]    4    0    0    2    0
#> [4,]    0    4    1    0    1
#> [5,]    0    0    0    6    0
#> 
#> , , 3
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0   10    0
#> [2,]    0    3    6    0    2
#> [3,]    0   16   -6    0    1
#> [4,]   10    0    0    1    0
#> [5,]    0   16    3    0   -8
#> 
#> , , 4
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    4   -3    0    5
#> [2,]    3    0    0    3    0
#> [3,]   -6    0    0   12    0
#> [4,]    0    4    6    0   -4
#> [5,]   30    0    0  -24    0
#> 
#> , , 5
#> 
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    0    0    0    0
#> [2,]    0   12   -9    0   -3
#> [3,]    0  -24   18    0    6
#> [4,]    0    0    0    0    0
#> [5,]    0  -24   18    0    6
#> 
#> 
#> $den
#> [1]   1   0 -14   0  49   0 -36   0

# Example 3:
iwish_psr(2, 1/2) # For quaternion Wishart distribution with k = 2
#> $c
#> , , 1
#> 
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> , , 2
#> 
#>      [,1] [,2]
#> [1,]  0.0  1.0
#> [2,]  0.5  0.5
#> 
#> 
#> $den
#> [1]  1.0  0.5 -0.5

Using the wishmom Package

Raymond Kan and Preston Liang

2024-08-25

Introduction

Mathematical Background

β-Wishart Distribution

Moments of Matrix-valued Functions of β-Wishart and Inverse β-Wishart Distributions

Main Functions in the Package

Moments of β-Wishart:

wishmom()

Arguments

Output

Examples

wishmom_sym()

Arguments

Output

Examples

Moments of Inverse β-Wishart:

iwishmom()

Arguments

Output

Examples

iwishmom_sym()

Arguments

Output

Examples

Auxiliary Functions

ip_desc()

Arguments

Output

Examples

dkmap()

Arguments

Output

Examples

denpoly()

Arguments

Output

Examples

qk_coeff()

Arguments

Output

Examples

wish_ps()

Arguments

Output

Examples

qkn_coeff()

Arguments

Output

Examples

iwish_ps()

Arguments

Output

Examples

qkn_coeffr()

Arguments

Output

Examples

iwish_psr()

Arguments

Output

Examples

References

$β$ -Wishart Distribution

Moments of Matrix-valued Functions of $β$ -Wishart and Inverse $β$ -Wishart Distributions

Moments of $β$ -Wishart:

Moments of Inverse $β$ -Wishart: