Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the number of elements beyond the ones actually observed

Eugenio Regazzini

TL;DR

This paper revisits de Finetti's continuous-time birth–death variant allowing unbounded negative jumps and places it within the modern CTMC framework on $\mathbb{Z}^+$. It derives the transition function $p^*_t$ via a generating-function approach, with a closed form involving the confluent hypergeometric function $\Phi(1,\vartheta+1,\vartheta z)$ where $\vartheta=\lambda/\mu$, and analyzes long-run behavior including the stationary distribution $\pi^*$ and ergodicity, along with explicit return-time formulas. It provides quantitative bounds on convergence to equilibrium and outlines practical uses for inference on unseen population elements from observed negative jumps, including a moment-based estimator for $\vartheta=\lambda/\mu$ and a Bayesian perspective. The framework supports inference on unseen abundance or richness in ecological and migratory contexts, and lays groundwork for likelihood-based and Bayesian estimation under partial observation of jumps.

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.

On the number of elements beyond the ones actually observed

TL;DR

This paper revisits de Finetti's continuous-time birth–death variant allowing unbounded negative jumps and places it within the modern CTMC framework on . It derives the transition function via a generating-function approach, with a closed form involving the confluent hypergeometric function where , and analyzes long-run behavior including the stationary distribution and ergodicity, along with explicit return-time formulas. It provides quantitative bounds on convergence to equilibrium and outlines practical uses for inference on unseen population elements from observed negative jumps, including a moment-based estimator for and a Bayesian perspective. The framework supports inference on unseen abundance or richness in ecological and migratory contexts, and lays groundwork for likelihood-based and Bayesian estimation under partial observation of jumps.

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.
Paper Structure (16 sections, 144 equations, 2 figures)

This paper contains 16 sections, 144 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of the process
  3. Expression of the transition function
  4. Equation of the generating function of the transition probability
  5. Long-time properties of the process
  6. Bounds on the error in approximating the equilibrium
  7. Outline of potential practical uses of the model
  8. Statistical estimation of the unknown probability intensities
  9. Definition of the estimator
  10. Asymptotic properties of the estimator
  11. Brief mention to the estimation of the rate of negative jumps
  12. Final remarks from a Bayesian perspective
  13. Expectation, variance and other moments of the distribution $\varphi_\vartheta$
  14. Extension of $L$ and its connection with the incomplete gamma function
  15. Evaluation of $L(\vartheta)$ for small and moderate values of $\vartheta$
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Graphs of the functions $L$, $L'$ and $L"$.
  • Figure 2: Graphs of $L$, $L'$, $L"$ and their asymptotic expansions for $\vartheta\to+\infty$, truncated at $n=5$, according to \ref{['eq-a4-1']}.

Theorems & Definitions (8)

  • proof : Proof of (\ref{['th-A']})
  • proof : Proof of (\ref{['th-F']})
  • Remark
  • proof : Proof of (\ref{['th-G']})
  • Remark
  • proof : Proof of (\ref{['th-J']})
  • proof : Proof of (\ref{['th-K']})
  • proof : Proof of (\ref{['th-L']})