单位尺度双重奇异Beta Type II系综的最大有限根
Largest Finite Root of Identity-Scale Doubly Singular Beta Type II Ensembles
AI总结：
本文针对高维中双重奇异Beta Type II系综，证明其最大有限根恒等于非奇异q维Roy统计量，从而可直接用经典算法计算有限样本分布。
摘要：
Wishart比率和矩阵变量Beta系综的经典最大根分布通常是在分母Wishart矩阵非奇异的情况下提出的。然而，在许多高维设定中，环境维度$p$超过两个Wishart自由度，因此相应的Beta系综是双重奇异的，通常的矩阵乘积$A^{-1}B$未定义。我们考虑独立中心Wishart矩阵$A\sim W_p(m,I_p)$和$B\sim W_p(q,I_p)$在$p>m\ge q$的范围内。对于对$(B,A)$的有限广义根，或等价地，$BA^+$的非零特征值，我们证明了精确恒等式\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*} 这里$W_d(r,I_d)$表示具有$r$自由度和单位尺度的$d\times d$中心Wishart矩阵。因此，单位尺度双重奇异Beta Type II最大根问题精确地简化为非奇异的$q$维Roy统计量，使其有限样本CDF可直接由经典最大根算法计算。
英文摘要：
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*} λ_{\max} \stackrel{d}{=} λ_{\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.
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