ON APPROXIMATING THE DISTRIBUTIONS OF RATIOS AND DIFFERENCES OF NONCENTRAL QUADRATIC FORMS IN NORMAL VECTORS

Abstract

The distribution of positive definite quadratic forms in normal random vectors
is first approximated by generalized gamma and Pearson-type density functions.
The distribution of indefinite quadratic forms is then obtained from their rep-
resentation in terms of the difference of two positive definite quadratic forms.
In the case of the Pearson-type approximant, explicit representations are ob-
tained for the density and distribution functions of an indefinite quadratic form.
A moment-based technique whereby the initial approximations are adjusted by
means of polynomials is being introduced. A detailed algorithm describing the
steps involved in the methodology advocated herein is provided as well. It is also
explained that the distributional results apply to the ratios of certain quadratic
forms. Two numerical examples are presented: the first involves an indefinite
quadratic form while the second approximates the distribution of the Durbin-
Watson statistic, which is shown to be expressible as a ratio of quadratic forms.