On Approximating the Distributions of Ratios and Differences of Noncentral Quadratic Forms in Normal Vectors (original) (raw)

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 representation in terms of the difference of two positive definite quadratic forms. In the case of the Pearson-type approximant, explicit representations are obtained 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.