Notes on the distribution of quadratic forms in singular normal variables

Abstract

Conditions for a quadratic form in singular normal variables to follow a χ2 distribution are obtained, correcting and extending previous results by Good (1969). Connexions with the noncentral case and independence of two quadratic forms are also given extending earlier results by Ogasawara & Takahashi (1951). Finally, a further generalization of the well-known theorem of Cochran (1934) is presented.