Let $S(b) = \Sigma r\sigma r^2Xr^2(nr,br^2)$ be a positive linear combination of independent noncentral chi-square random variables. This note derives two representations for the tail probabilities P[S(b) >x], a Taylor series in the noncentrality parameters and a limiting form of this series for large x. An application of the latter result to statistical tests of Cramer-von Mises type is discussed.
"Tail Probabilities of Noncentral Quadratic Forms." Ann. Statist. 3 (4) 969 - 974, July, 1975. https://doi.org/10.1214/aos/1176343199