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 normal 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推导出多元正态总体中样本协方差矩阵嵌入主子式联合矩的显著公式，并将其用于计算多元线性回归相关应用中样本统计量的矩。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分解加以推广。本文利用迭代Schur补概念给出Wilks结果的新证明，从而绕过了Bartlett分解。此外，Wilks关于Wishart随机矩阵不交主子式联合矩评估的未解决问题，与高斯乘积不等式猜想相关联。