On approximating the distribution of indefinite quadratic expressions in singular normal vectors

Abstract

General representations of quadratic forms and quadratic expressions in singular normal vectors are given in terms of the difference of two positive definite quadratic forms and an independently
distributed linear combination of normal random variables. Up to now, only special cases have been treated in the statistical literature. The densities of the quadratic forms are then approximated with gamma and generalized gamma density functions. A moment-based technique whereby the initial approximations are adjusted by means of polynomials is presented. Closed form and integral formulae are provided for the approximate density functions of the quadratic forms and quadratic expressions. A detailed step-by-step algorithm for implementing the proposed density approximation technique is also provided. Two numerical examples illustrate the methodology.