This paper presents likelihood-based inference methods for the family of univariate gamma-normal distributions GN({\alpha}, r, {\mu}, {\sigma}^2 ) that result from summing independent gamma({\alpha}, r) and N({\mu}, {\sigma}^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.

翻译：本文针对单变量伽马正态分布族GN(α, r, μ, σ²)提出了基于似然的推断方法，该分布由独立伽马分布Γ(α, r)与正态分布N(μ, σ²)随机变量之和生成。首先，利用抛物柱函数以紧凑形式给出了伽马正态变量的概率密度函数及其关键性质。随后，我们推导了最大似然得分方程与费希尔信息矩阵的解析表达式，并讨论了伽马正态分布的推断方法。鉴于其两个构成分布的广泛使用，伽马正态分布可成为适用于多种场景的通用工具。特别地，我们重点讨论了作为特例获得的两种分布——指数正态分布与卡方正态分布（或称过离散卡方分布），这两种分布在众多统计应用中具有重要地位。