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 α1χd12+β1, where the unknown parameters <i>α</i>₁, <i>β</i>₁, and <i>d</i>₁ 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 αχd2+β+σξ, where <i>ξ</i> is a standard normal random variable, and the unknown parameters <i>α</i>, <i>β</i>, <i>σ</i>, and <i>d</i> 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.