A novel method for approximating the distribution of chi-squared-type mixtures

In order to approximate the distribution of chi-squared-type mixtures, Zhang (2005) proposed to use a chi-squared-type random variable of the form alpha 1 chi d12+beta 1, where the unknown parameters alpha 1, beta 1, and d1 are determined by matching the first three cumulants. In this article, we propose a novel method to approximate the distribution of chi-squared-type mixtures by the distribution of a random variable in the form alpha chi d2+beta+sigma xi, where xi is a standard normal random variable, and the unknown parameters alpha, beta, sigma, and d are determined by matching the first four cumulants. The approximating error bound on the distribution functions of the new method approximation is established. The numerical results show that our proposed method can has fewer error bound than some existed methods in some examples.