Distribution of Quadratic Forms and Ratios of Quadratic Forms

Abstract

Let the random variable $X = (X_1, X_2, \cdots, X_n)$ have the probability density $p(x) = \frac{\det^{\frac{1}{2}} \Omega}{(2\pi)^{n/2}} e^{-\frac{1}{2}x\Omega x'}$ where $x\Omega x'$ is positive definite. The present article solves, by means of Laguerian expansions, the problem of finding the distribution of any nonnegative quadratic form $XPX'$. If the semimoments (defined below) are known, it also solves, by means of Laguerrian expansions the problems of finding the distribution of any indefinite quadratic form, and the distribution of the ratio of any indefinite quadratic form to any nonnegative quadratic form. For an outline of the procedure, see Section 2. If the distribution of the indefinite form is symmetric, the semimoments are easily found, but often, especially for the technique described below for ratios, the semimoments are difficult to obtain. In view of this, a new system of orthogonal polynomials is proposed, which is analogous to the Laguerre system, but which obviates the need of semimoments.