The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variable (IV) regression, but there has been very little work on its properties. In this paper, we first provide an expression for the expectation of the inverse of a noncentral Wishart matrix, and then go on to do the same for a number of scalar-valued functions of the inverse. The main result is obtained by exploiting simple but powerful group-equivariance properties of the expectation map involved. Subsequent results exploit the consequences of other invariance properties.

中文翻译：

---

非中心 WISHART 矩阵的逆矩阵的性质

非中心 Wishart 矩阵的逆矩阵出现在多元统计工作的各种情况下，包括工具变量 (IV) 回归，但对其属性的研究很少。在本文中，我们首先为非中心 Wishart 矩阵的逆矩阵的期望提供一个表达式，然后继续为逆矩阵的多个标量值函数做同样的事情。主要结果是通过利用所涉及的期望图的简单但强大的组等方差特性获得的。随后的结果利用了其他不变性属性的结果。