Exact Largest Eigenvalue Distribution for Doubly Singular Beta Ensemble

Abstract

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