Dispersion indexes with respect to the Poisson and binomial distributions are widely used to assess the conformity of the underlying distribution from an observed sample of the count with one or the other of these theoretical distributions. Recently, the exponential variation index has been proposed as an extension to nonnegative continuous data. This paper aims to gather to study the unified definition of these indexes with respect to the relative variability of a nonnegative natural exponential family of distributions through its variance function. We establish the strong consistency of the plug-in estimators of the indexes as well as their asymptotic normalities. Since the exact distributions of the estimators are not available in closed form, we consider the test of hypothesis relying on these estimators as test statistics with their asymptotic distributions. Simulation studies globally suggest good behaviours of these tests of hypothesis procedures. Applicable examples are analysed, including the lesser-known references such as negative binomial and inverse Gaussian, and improving the very usual case of the Poisson dispersion index. Concluding remarks are made with suggestions of possible extensions.

中文翻译：

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统一相对离散和相对变异指标的检验统计量的渐近正态性

关于泊松分布和二项分布的分散指数被广泛用于评估观察到的计数样本的基本分布与这些理论分布中的一种或另一种的一致性。最近，已提出指数变化指数作为对非负连续数据的扩展。本文旨在通过其方差函数收集研究这些指标关于非负自然指数分布族的相对变异性的统一定义。我们建立了索引的插件估计量及其渐近正态性的强一致性。由于估计量的精确分布不能以封闭形式获得，我们将依赖这些估计量的假设检验视为具有渐近分布的检验统计量。全球模拟研究表明这些假设程序检验的良好行为。分析了适用的示例，包括鲜为人知的参考文献，例如负二项式和逆高斯，并改进了泊松色散指数的非常常见的情况。总结性评论附有可能的扩展建议。