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.
From: Stepan Grinek [view email]
[v1]
Mon, 6 May 2019 00:37:42 UTC (247 KB)
[v2]
Tue, 7 May 2019 19:19:28 UTC (247 KB)
[v3]
Sat, 4 Jan 2020 17:10:51 UTC (247 KB)
[v4]
Fri, 4 Sep 2026 18:59:15 UTC (80 KB)
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