Distribution of Definite and of Indefinite Quadratic Forms

Abstract

A previous paper [1] has given a method of approximating the distribution of a quadratic form in normally distributed variables by means of convergent Laguerrian expansions. In the case of an indefinite quadratic form, however, the method was restrictive in that it might be difficult to obtain the semi-moments required in computing the coefficients of the expansion. The present article circumvents this difficulty for positive values of the argument of the distribution function, when the number of positive or the number of negative eigenvalues is even, and also yields convergent expansions for the distribution function involving Laguerre polynomials. The proposed method has the further advantage that no moments or semi-moments need be calculated.