Tail areas of linear combinations of chi-squares and non-central chi-squares

This paper uses a uniform saddlepoint approximation to obtain extreme tail areas for a linear combination of independent chi-square random variables. The approximation uses a transformation of the tail area coming from the Fourier inversion of the characteristic function. The transformation yields a quadratic in the exponent. A Taylor's expansion of the Jacobian leads to the approximation. The approximation is shown to be easy to compute and very accurate in the extreme tails.