Largest Finite Root of Identity-Scale Doubly Singular Beta Type II Ensembles

Abstract

Classical largest-root distributions for Wishart ratios and matrix-variate beta ensembles are usually formulated when the denominator Wishart matrix is nonsingular. In many high-dimensional settings, however, the ambient dimension $p$ exceeds both Wishart degrees of freedom, so the corresponding beta ensemble is doubly singular and the usual matrix product $A^{-1}B$ is not defined. We consider independent central Wishart matrices $A\sim W_p(m,I_p)$ and $B\sim W_p(q,I_p)$ in the regime $p>m\ge q$. For the finite generalized roots of the pair $(B,A)$ or, equivalently, the nonzero eigenvalues of $BA^+$, we prove the exact identity \begin{equation*} \lambda_{\max} \stackrel{d}{=} \lambda_{\max}\left\{W_q(m,I_q)W_q(p-m+q,I_q)^{-1}\right\}. \end{equation*} Here $W_d(r,I_d)$ denotes a $d\times d$ central Wishart matrix with $r$ degrees of freedom and identity scale. Thus, the identity-scale doubly singular beta type II largest-root problem reduces exactly to a nonsingular $q$-dimensional Roy statistic, making its finite-sample CDF directly accessible to classical largest-root algorithms.