A Gaussian Approximation to the Distribution of a Definite Quadratic Form

Abstract Let Qk = Σk j-1 cj(xj + aj)2 be a definite quadratic form in independent standardized Gaussian variables, xj, EQk = 01. The normalizing transformation (Qk/01)h is investigated, where h is determined by the first three moments of Qk. A new Gaussian approximation to the noncentral chi-square (x2) distribution is found for which the coefficient of skewness is smaller than a cube root transformation in the literature. Our transformation specializes to the cube root transformation of Wilson and Hilferty for the central x2 distribution. The approximation is simple to apply and compares well with other approximations in a number of cases studied numerically.