The Distribution of Quadratic Forms in Non-Normal Variables and an Application to the Variance Ratio
A. I. Knuri, Irving John Good · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 1977
Summary Good (1968a, b) using non-rigorous arguments, gave an expression for the characteristic function C.F. (x‘A x, t) of a quadratic form in a non-normal random column vector x whose characteristic function is known, where A is positive semi-definite. A general formula for C.F. (x‘A x, t) is proved here, which coincides with Good's expression when x has a real stable characteristic function. In the general formula, A need not be positive semi-definite and this greater generality is of value for finding the distribution of the ratio of two quadratic forms in non-normal variables. The theory is used for finding approximately the distribution of a ratio of quadratic forms in a three-dimensional random vector x having an even stable distribution with exponent 3/2. The procedure of regarding x as normal, combined with the Cornish–Fisher expansion, would have been far from robust in this example.