  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.
