Notes on a theorem of Cochran

Extract

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 x₁, …, x_n are independently distributed standard normal deviates, and q₁, …, q_k are k real quadratic forms in the x_i with ranks n₁, …, n_k respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q₁ …, q_k are independently distributed in χ² forms with n₁, …, n_k degrees of freedom is that Σn_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.