\(\newcommand{\etr}{\textrm{etr}}\)
Two definitions of the matrix variate Beta type I distribution were proposed. We will denote them by \(\mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)\) and \(\mathcal{B}I_p^{(2)}(a,b,\Theta_1,\Theta_2)\), where \(\Theta_1\) and \(\Theta_2\) are the noncentrality parameters. Take two independent Wishart random matrices \(W_1 \sim \mathcal{W}_p(2a, I_p, \Theta_1)\) and \(W_2 \sim \mathcal{W}_p(2b, I_p, \Theta_2)\). Then \(\mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)\) is the distribution of \[ U_1 = {(W_1+W_2)}^{-\frac12}W_1{(W_1+W_2)}^{-\frac12}, \] while \(\mathcal{B}I_p^{(2)}(a,b,\Theta_1,\Theta_2)\) is the distribution of \[ U_2 = W_1^\frac12{(W_1+W_2)}^{-1}W_1^\frac12. \] The condition \(a+b > \frac12(p-1)\) is required in order for \(W_1+W_2\) to be invertible.
In the central case, i.e. when both \(\Theta_1\) and \(\Theta_2\) are the null matrices, these two distributions are the same. More generally, as we will see, they are the same when \(\Theta_1\) and \(\Theta_2\) are scalar.
Similarly, two definitions of the matrix variate Beta type II distribution were proposed. We will denote them by \(\mathcal{B}II_p^{(1)}(a,b,\Theta_1,\Theta_2)\) and \(\mathcal{B}II_p^{(2)}(a,b,\Theta_1,\Theta_2)\). The first one is the distribution of \[ V_1 = W_2^{-\frac12} W_1 W_2^{-\frac12}, \] while the second one is the distribution of \[ V_2 = W_1^\frac12 {W_2}^{-1} W_1^\frac12. \] The condition \(b > \frac12(p-1)\) is required in order for \(W_2\) to be invertible.
Similarly to the type I, these two distributions are the same in the central case, and more generally when \(\Theta_1\) and \(\Theta_2\) are scalar.
Under the second definition, the Beta type I distribution is related to the Beta type II distribution by \(U_2 \sim V_2{(I_p+V_2)}^{-1}\).
The densities of the matrix Beta distributions involve the hypergeometric function of matrix argument \({}_0\!F_1\). We will use the property \({}_0\!F_1(\alpha, AB)={}_0\!F_1(\alpha, BA)\) (to simplify the densities when \(\Theta_1\) or \(\Theta_2\) are scalar).
Recall that \[ U_1 = {(W_1+W_2)}^{-\frac12} W_1 {(W_1+W_2)}^{-\frac12}. \] It is clear from this definition that \[ I_p - U_1 \sim \mathcal{B}I_p^{(1)}(b, a, \Theta_2, \Theta_1). \] The density of \(U_1\) is \[ \begin{aligned} \mathcal{B}I_p^{(1)}(U \mid a, b, \Theta_1, \Theta_2) \propto \, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr\left(-S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S^{\frac12} U S^\frac12\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S^{\frac12}(I_p-U)S^\frac12\right) \mathrm{d}S. \end{aligned} \]
If \(\Theta_1\) and \(\Theta_2\) are scalar: \[ \begin{aligned} \mathcal{B}I_p^{(1)}(U \mid a, b, \theta_1 I_p, \theta_2 I_p) \propto\, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr\left(-S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{\theta_1}{2}SU\right) {}_0\!F_1\left(b, \frac{\theta_2}{2}S(I_p-U)\right) \mathrm{d}S. \end{aligned} \]
Recall that \[ U_2 = W_1^\frac12{(W_1+W_2)}^{-1}W_1^\frac12. \]
The density of \(U_2\) is \[ \begin{aligned} \mathcal{B}I_p^{(2)}(U \mid a, b, \Theta_1, \Theta_2) \propto \, & {\det(U)}^{-b-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr\left(-S U^{-1}\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2 S^\frac12 U^{-1}(I_p-U)S^{\frac12}\right)\mathrm{d}S. \end{aligned} \]
If \(\Theta_1\) and \(\Theta_2\) are scalar, it is equal to the density of \(\mathcal{B}I_p^{(1)}(a, b, \Theta_1, \Theta_2)\).
More generally, we give the density of \[ V_1 = {(W_2^{-\frac12})}' W_1 W_2^{-\frac12} \] where \(W_2^{\frac12}{(W_2^{\frac12})}' = W_2.\)
This density is \[ \begin{aligned} \mathcal{B}II_p^{(1)}(V \mid a, b, \Theta_1, \Theta_2) \propto \, & {\det(V)}^{a-\frac12(p+1)} \\ & \int_{S>0} \etr\bigl(-(I_p+V)S\bigr) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1{(S^{\frac12})}' V S^{\frac12}\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S\right) \mathrm{d}S. \end{aligned} \]
If \(\Theta_1\) is scalar, the distribution does not depend on the choice of \(W_2^\frac12\).
Recall that \[ V_2 = W_1^{\frac12} W_2^{-1} {(W_1^{\frac12})}'. \] It is clear from the definitions that \[ V_2^{-1} \sim \mathcal{B}II_p^{(1)}(b,a,\Theta_2,\Theta_1). \]
\[ \begin{aligned} \mathcal{B}II_p^{(2)}(V \mid a, b, \Theta_1, \Theta_2) \propto \, & {\det(V)}^{-b-\frac12(p+1)} \\ & \int_{S >0} \etr\left(-S(I_p+V^{-1})\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2{(S^\frac12)}' V^{-1} S^\frac12\right) \mathrm{d}S. \end{aligned} \]
If \(\Theta_2\) is scalar, the distribution does not depend on the choice of \(W_1^\frac12\).
If \(\Theta_1\) and \(\Theta_2\) are scalar, this is the same distribution as \(\mathcal{B}II_p^{(1)}(a,b,\Theta_1,\Theta_2)\).
If we take \(W_1^{\frac12}\) the symmetric square root of \(W_1\), then \(V_2{(I_p+V_2)}^{-1} \sim \mathcal{B}I_2(a,b,\Theta_1,\Theta_2)\).
Recall that \(\mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)\) is the distribution of \[ U = {(W_1+W_2)}^{-\frac12}W_1{(W_1+W_2)}^{-\frac12}. \]
The joint density of \(S_1\) and \(S_2\) is \[ C \, \etr\left(-\frac12 S_1\right)\etr\left(-\frac12 S_2\right) {\det(S_1)}^{a-\frac12(p+1)} {\det(S_2)}^{b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S_1\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S_2\right). \] Using the transformation \(S_1+S_2=S\) and \(S_1 = S^{\frac12} U S^\frac12\), with Jacobian \(J(S_1, S_2 \rightarrow U, S) = {\det(S)}^{\frac12(p+1)}\), we get the pdf of \((U,S)\) as \[ \begin{aligned} C\, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \etr\left(-\frac12 S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S^{\frac12} U S^\frac12\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S^{\frac12}(I_p-U) S^\frac12\right). \end{aligned} \] Thus the density of \(U_1\) is \[ \begin{aligned} C\, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr\left(-\frac12 S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S^{\frac12}U{(S^\frac12)}'\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S^{\frac12}(I_p-U)S^\frac12\right) \mathrm{d}S \\ = C\,& {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr(-S) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S^{\frac12} U S^\frac12\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S^{\frac12}(I_p-U)S^\frac12\right) \mathrm{d}S. \end{aligned} \]
Recall that \(\mathcal{B}I_p^{(2)}(a,b,\Theta_1,\Theta_2)\) is the distribution of \[ U = W_1^\frac12{(W_1+W_2)}^{-1}W_1^\frac12. \]
Using the transformation \(S_1+S_2 = S_1^{\frac12}U^{-1}S_1^{\frac12}\), with Jacobian \(J(S_1, S_2 \rightarrow U, S_1) = {\det(S_1)}^{\frac12(p+1)} {\det(U)}^{-(p+1)}\), we get the pdf of \((U,W_1)\) as \[ \begin{aligned} C \, & {\det(S_1)}^{a+b-\frac12(p+1)} {\det(U)}^{-(p+1)} {\det(U^{-1}-I_p)}^{b-\frac12(p+1)} \\ & \etr\left(-\frac12 S_1U^{-1}\right) {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S_1\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S_1^{\frac12}(U^{-1}-I_p)S_1^\frac12\right) \\ = C \, & {\det(S_1)}^{a+b-\frac12(p+1)} {\det(U)}^{-b-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \etr\left(-\frac12 S_1U^{-1}\right) {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S_1\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S_1^{\frac12}U^{-1}(I_p-U)S_1^\frac12\right). \end{aligned} \] Thus the density of \(U\) is \[ \begin{aligned} C\, & {\det(U)}^{-b-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S_1>0} \etr\left(-\frac12 S_1U^{-1}\right) {\det(S_1)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S_1\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S_1^{\frac12}U^{-1}(I_p-U)S_1^\frac12\right) \mathrm{d}S_1\\ = C\, & {\det\bigl(U{(I_p-U)}^{-1}\bigr)}^{-b-\frac12(p+1)} {\det(I_p-U)}^{-(p+1)} \\ & \int_{S>0} \etr\left(-S U^{-1}\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S^{\frac12}U^{-1}(I_p-U)S^\frac12\right) \mathrm{d}S. \end{aligned} \] Let’s derive the density of \(U{(I_p-U)}^{-1}\). Using the transformation \(U = V{(I_p+V)}^{-1}\) with Jacobian \(J(U \rightarrow V) = {\det(I_p+V)}^{-(p+1)}\), we get the density of \(V\) as \[ \begin{aligned} C\, & {\det(V)}^{-b-\frac12(p+1)} \\ & \int_{S>0} \etr\bigl(-S (I_p+V^{-1})\bigr) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S^{\frac12}V^{-1}S^\frac12\right) \mathrm{d}S. \end{aligned} \] This is the density of \(\mathcal{B}II_p^{(2)}(a,b,\Theta_1,\Theta_2)\).
Recall that \(\mathcal{B}II_p^{(1)}(a,b,\Theta_1,\Theta_2)\) is the distribution of \[ V = {(W_2^{-\frac12})}' W_1 W_2^{-\frac12}. \]
Transforming \(S_1 = {(S_2^{\frac12})}' V S_2^{-\frac12}\) with Jacobian \(J(S_1, S_2 \rightarrow V, S_2) = {\det(S_2)}^{\frac12(p+1)}\), we get the density of \((V,S_2)\) as \[ \begin{aligned} C\, & {\det(V)}^{a-\frac12(p+1)} {\det(S_2)}^{a+b-\frac12(p+1)} \\ & \etr\left(-\frac12 (I_p+V)S_2\right) {}_0\!F_1\left(a, \frac{1}{4}\Theta_1 {(S_2^{\frac12})}' V S_2^{-\frac12}\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2S_2\right). \end{aligned} \] Thus, the density of \(V\) is \[ \begin{aligned} C\, & {\det(V)}^{a-\frac12(p+1)} \\ & \int_{S>0} \etr\bigl(-(I_p+V)S\bigr) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1{(S^{\frac12})}' V S^{-\frac12}\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S\right) \mathrm{d}S. \end{aligned} \]
Recall that \(\mathcal{B}II_p^{(2)}(a,b,\Theta_1,\Theta_2)\) is the distribution of \[ V = W_1^{\frac12} W_2^{-1} {(W_1^{\frac12})}'. \]
Transforming \(S_2 = {(S_1^\frac12)}' V^{-1} S_1^\frac12\) with Jacobian \(J(S_1, S_2 \rightarrow V, S_1) = {\det(S_1)}^{\frac12(p+1)}{\det(V)}^{-(p+1)}\), we get the density of \((V,S_1)\) as \[ \begin{aligned} C\, & {\det(S_1)}^{a+b-\frac12(p+1)} {\det(V)}^{-b-\frac12(p+1)} \\ & \etr\left(-\frac12 S_1\right)\etr\left(-\frac12 S_1V^{-1}\right) {}_0\!F_1\left(a, \frac{1}{4}\Theta_1S_1\right) {}_0\!F_1\left(b, \frac{1}{4}\Theta_2{(S_1^\frac12)}' V^{-1} S_1^\frac12\right). \end{aligned} \] Thus the density of \(V\) is \[ \begin{aligned} C\, & {\det(V)}^{-b-\frac12(p+1)} \\ & \int_{S >0} \etr\bigl(-(I_p+V^{-1})S\bigr) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2{(S^\frac12)}' V^{-1} S^\frac12\right) \mathrm{d}S. \end{aligned} \] In the case when \(\Theta_2\) is scalar, \[ {}_0\!F_1\left(b, \frac{1}{2}\Theta_2{(S^\frac12)}' V^{-1} S^\frac12\right) = {}_0\!F_1\left(b, \frac{1}{2}\Theta_2 S V^{-1}\right). \] Doing the change of variables \(R = S V^{-1}\) in the integral, we get the density of \(V\) as \[ \begin{aligned} C\, & {\det(V)}^{a-\frac12(p+1)} \\ & \int_{R >0} \etr\bigl(-(I_p+V)R\bigr) {\det(R)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1 R V\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2 R\right) \mathrm{d}R. \end{aligned} \] If in addition, \(\Theta_1\) is scalar, this is the density of \(\mathcal{B}II_p^{(1)}(a,b,\Theta_1,\Theta_2)\).
One has \({}_0\!F_1(\alpha, \mathbf{0}) = 1\). Thus, when \(\Theta_1\) or \(\Theta_2\) is the null matrix, we get integrals like \[ \int_{S>0} \etr(-ZS) {\det(S)}^{\alpha-\frac12(p+1)} {}_0\!F_1\left(\beta, \frac{1}{2}\Theta S T \right) \mathrm{d}S, \] for example in the density of \(\mathcal{B}I_p^{(2)}(a,b,\Theta_1,\Theta_2)\) when \(\Theta_2\) is the null matrix, or in the density of \(\mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)\) when \(\Theta_1\) is the null matrix and \(\Theta_2\) is scalar.
This integral is equal to \[ \Gamma_p(\alpha){\det(Z)}^{-\alpha} {}_1\!F_1\left(\alpha, \beta, \frac{1}{2}\Theta Z^{-1} T \right). \]