On the number of elements beyond the ones actually observed

Eugenio Regazzini
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Abstract

In this work, a variant of the birth and death chain with constant intensities, originally introduced by Bruno de Finetti way back in 1957, is revisited. This fact is also underlined by the choice of the title, which is clearly a literal translation of the original one. Characteristic of the variant is that it allows negative jumps of any magnitude. And this, as explained in the paper, might be useful in offering some insight into the issue, arising in numerous situations, of inferring the number of the undetected elements of a given population. One thinks, for example, of problems concerning abundance or richness of species. The author's purpose is twofold: to align the original de Finetti's construction with the modern, well-established theory of the continuous-time Markov chains with discrete state space and show how it could be used to make probabilistic previsions on the number of the unseen elements of a population. With the aim of enhancing the possible practical applications of the model, one discusses the statistical point estimation of the rates which characterize its infinitesimal description.
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关于实际观测之外的元素数量
在这部作品中,我们重新审视了布鲁诺-德-菲内蒂(Bruno de Finetti)早在 1957 年就提出的具有恒定强度的生死链变体。标题的选择也凸显了这一事实,它显然是对原标题的直译。该变式的特点是允许任何大小的负跳跃。正如论文中解释的那样,这可能有助于深入了解在许多情况下出现的问题,即推断特定种群中未检测到的元素的数量。例如,我们会想到有关物种丰度或丰富度的问题。作者的目的有两个:将德菲内蒂的原始结构与现代成熟的离散空间连续时间马尔可夫链理论相结合,并说明如何利用它来对种群中未发现元素的数量进行概率预测。为了加强该模型可能的实际应用,我们讨论了该模型无穷小描述的特征率的统计点估计。
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