In 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but little-known moment results were extended in 1963 by the Australian statistician A. Graham Constantine using Bartlett's decomposition. In this note, a new proof of Wilks' results is derived using the concept of iterated Schur complements, thereby bypassing Bartlett's decomposition. Furthermore, Wilks' open problem of evaluating joint moments of disjoint principal minors of Wishart random matrices is related to the Gaussian product inequality conjecture.

翻译：1934年，美国统计学家Samuel S. Wilks针对多元高斯总体中样本协方差矩阵的嵌入主子式联合矩，推导出了卓越的公式，并将其用于计算多元线性回归相关应用中各类样本统计量的矩。这些重要但鲜为人知的矩结果于1963年由澳大利亚统计学家A. Graham Constantine利用Bartlett分解进行了推广。本文通过迭代Schur补的概念，导出了Wilks结果的一个新证明，从而绕过了Bartlett分解。此外，Wilks关于评估Wishart随机矩阵不相交主子式联合矩的开放问题，与高斯乘积不等式猜想相关联。