We study an iterative discrete information production process (IPP) 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 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. A case study of family income data in nine countries shows a very good fit of our model. Moreover, we show that asymptotically, we obtain all, and only, Lorenz curves generated by a new, intuitive parametrisation of the finite-mean Generalised Pareto Distribution (GPD) that unifies three other families, namely: the Pareto Type II, exponential, and scaled beta ones. The family is not only ordered with respect to the parameter G, but also, thanks to our derivations, has a nice underlying interpretation. Our result may thus shed new light on the genesis of this family of distributions.

翻译：我们研究了一个迭代离散信息生产过程（IPP），在该过程中，基于简单的仿射变换，我们可以通过新元素扩展有序归一化向量，同时保持由基尼指数G衡量的预定义不平等水平。然后，我们推导了相应向量的洛伦兹曲线族，并证明该族在样本量和扮演不确定性参数角色的G方面具有随机序性质。对九个国家家庭收入数据的案例研究表明，我们的模型拟合效果非常好。此外，我们证明，在渐近情况下，我们能够获得且仅能获得由有限均值广义帕累托分布（GPD）的一种新的直观参数化所生成的所有洛伦兹曲线，这种参数化统一了另外三个分布族，即：帕累托II型分布、指数分布和缩放贝塔分布。该族不仅相对于参数G有序，而且得益于我们的推导，它还具有一个良好的潜在解释。因此，我们的结果可能为这类分布族的起源提供新的见解。