We introduce an iterative discrete information production process where we can extend ordered normalised vectors by new elements based on a simple affine transformation, while preserving the predefined level of inequality, G, as measured by the Gini index. Then, we derive the family of empirical Lorenz curves of the corresponding vectors and prove that it is stochastically ordered with respect to both the sample size and G which plays the role of the uncertainty parameter. We prove that asymptotically, we obtain all, and only, Lorenz curves generated by a new, intuitive parametrisation of the finite-mean Pickands' Generalised Pareto Distribution (GPD) that unifies three other families, namely: the Pareto Type II, exponential, and scaled beta ones. The family is not only totally ordered with respect to the parameter G, but also, thanks to our derivations, has a nice underlying interpretation. Our result may thus shed a new light on the genesis of this family of distributions. Our model fits bibliometric, informetric, socioeconomic, and environmental data reasonably well. It is quite user-friendly for it only depends on the sample size and its Gini index.

翻译：我们提出了一种迭代离散信息生产过程，在该过程中，基于简单的仿射变换，我们能够通过新元素扩展有序归一化向量，同时保持由基尼指数G测度的预定义不平等水平。随后，我们推导了对应向量的经验洛伦兹曲线族，并证明该曲线族关于样本量和作为不确定性参数的G均具有随机序性质。我们证明，渐近地，我们将获得且仅获得由有限均值Pickands广义帕累托分布（GPD）的一种新的直观参数化生成的洛伦兹曲线，该参数化统一了另外三个分布族：帕累托类型II族、指数族和缩放贝塔族。该分布族不仅关于参数G完全有序，而且通过我们的推导还具有良好的底层解释。因此，我们的结果可能为这一分布族的起源提供新的见解。我们的模型能较好地拟合文献计量学、信息计量学、社会经济和环境数据。该模型仅依赖于样本量及其基尼指数，因而非常便于使用。