Exact Largest Eigenvalue Distribution for Doubly Singular Beta Ensemble

In [1] beta type I and II doubly singular distributions were introduced and their densities and the joint densities of nonzero eigenvalues were derived. We found simple formula to compute largest root distribution for doubly singular beta ensemble in case $\Sigma=I$. Distribution is presented in terms of existing expression for CDF of Roy's statistic: $\lambda_{\max} \sim \max \eig \left\{ W_q(I, m)W_q(I, p-m+q)^{-1}\right\},$ where $W_p(I, n)$ is Wishart distribution with $p$ dimensions, $n$ degrees of freedom and identity scale matrix, $p \geq m$.