Notes on a theorem of Cochran

Abstract

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x 1 , …, x n are independently distributed standard normal deviates, and q 1 , …, q k are k real quadratic forms in the x i with ranks n 1 , …, n k respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q 1 …, q k are independently distributed in χ 2 forms with n 1 , …, n k degrees of freedom is that Σ n j = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each q j as a sum, involving n j squares of real linear forms in the x i ; it follows easily that the coefficients c i are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.