Distributions of Quadratic Forms

For independent chi-square variables x squared sub m and x squared sub n with m and n degrees of freedom, respectively, we consider the quadratic form where the positive ci are distinct. This paper gives exact finite expressions for the distribution of Q in terms of available functions such as the distribution function of chi-square random variables, modified Bessel Functions, Dawsons integral. These formulas are useful for checking the accuracy of approximations and tables of the distribution of Q and provide a simple alternative in their absence. For large m and n, reasonable approximations to the distribution of Q are available. For the general quadratic form Williams 1984 compares algorithms for truncations of infinite series expansions of the distribution.