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This paper presents likelihood-based inference methods for the family of univariate gamma-normal distributions GN (, r, , ²) that result from summing independent gamma (, r) and N (, ²) random variables. First, the probability density function of a gamma-normal variable is provided in compact form with the use of parabolic cylinder functions, along with key properties. We then provide analytic expressions for the maximum-likelihood score equations and the Fisher information matrix, and discuss inferential methods for the gamma-normal distribution. Given the widespread use of the two constituting distributions, the gamma-normal distribution is a general purpose tool for a variety of applications. In particular, we discuss two distributions that are obtained as special cases and that are featured in a variety of statistical applications: the exponential-normal distribution and the chi-squared-normal (or overdispersed chi-squared) distribution.
Bonamente et al. (Fri,) studied this question.