On a new transformation to normality

A Gaussian approximation to the distribution of the nonnegative random variable Y is developed using the Wilson and Hilferty (1931) approach. This approximation uses the symmetrizing transformation ((Y + b)/k₁)^h where k₁ is the first moment of Y and h and b are determined from the first three cumulants of Y. The approximation is illustrated in the case which Y is a non-central chi-square, where numerical evaluations indicate that the new transformation is an improvement over existing ones, especially for small values of k₁.