It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures. The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework. The main aim of the paper is to expose the mean value parametrization for possible statistical applications. The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s. In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation. Some classes of polynomial variance functions are revisited for illustration. The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution.

中文翻译：

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关于自然指数家庭的均值参数化的回顾

众所周知，任何自然指数族（NEF）的特征在于其均值域上的方差函数，通常比相应的生成概率测度简单得多。在某些统计理论中，平均值参数化似乎至关重要，例如在广义线性模型，指数弥散模型和贝叶斯框架中。本文的主要目的是为可能的统计应用公开平均值参数化。本文概述了NEF的均值参数化和方差函数的特征。特别是，它介绍了NEF的生成度量，拉普拉斯变换和方差函数之间存在的关系，以及有关平均值表示的一些补充结果。再次介绍了一些类别的多项式方差函数以进行说明。通过计算非负整数的概率，可以生成此类的相应NEF，并将其分布在Poisson过度分散的竞争者中，以提供均质的Poisson分布。