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：本文提出基于似然的推断方法，适用于单变量伽马-正态分布族GN(α, r, μ, σ²)，该分布由独立伽马(α, r)与正态N(μ, σ²)随机变量之和生成。首先，借助抛物柱面函数以紧凑形式给出伽马-正态变量的概率密度函数，并阐述其关键性质。继而推导极大似然得分方程与费舍尔信息矩阵的解析表达式，并讨论伽马-正态分布的推断方法。鉴于两个组成分布的广泛应用，伽马-正态分布成为多种应用场景中的通用工具。特别地，本文讨论两种作为特例获得且在各类统计应用中常见的分布：指数-正态分布与卡方-正态分布（或过度分散卡方分布）。