Approximations to Joint Distributions of Definite Quadratic Forms

Abstract

Multidimensional Wilson-Hilferty transformations support Gaussian approximations to certain joint distributions of quadratic forms in jointly Gaussian variates. Central and noncentral joint distributions are studied and applications are noted. Parameters of the approximating distributions are given up to terms of order o(v ⁻²) in the degrees of freedom v. Numerical studies validate using these approximations over a range of parameters for an essential subclass of the distributions studied, especially in upper tails.

Key Words:

• Distributions of quadratic forms
• Gaussian approximations
• Multivariate chi-squared and Rayleigh distributions
• Normalizing transformations