On Wilks' joint moment formulas for embedded principal minors of Wishart random matrices
arxiv(2024)
摘要
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.
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