On the number of elements beyond the ones actually observed

TL;DR

This paper revisits a birth-death Markov chain model allowing negative jumps, originally by de Finetti, to estimate the number of unseen elements in populations, with applications in ecology and related fields.

Contribution

It aligns de Finetti's original model with modern Markov chain theory and explores its use for predicting unobserved population elements.

Findings

Provides a modern probabilistic framework for de Finetti's model.

Discusses statistical estimation of model parameters.

Offers insights into unseen population size inference.

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.