Another view of sequential sampling in the birth process with immigration

TL;DR

This paper investigates properties of a continuous-time birth process with immigration model for counts-of-counts data, analyzing correlations, variances, and asymptotics, with applications to biological data and Fisher's multi-sampling problem.

Contribution

It introduces a continuous-time framework for counts-of-counts data in a birth process with immigration, simplifying asymptotic analysis and connecting to Chinese Restaurant Process samples.

Findings

Derived expected sample variance and asymptotics for sequential samples

Connected continuous-time model to Chinese Restaurant Process sampling

Reinterpreted Fisher's 1943 multi-sampling problem in this context

Abstract

Models of counts-of-counts data have been extensively used in the biological sciences, for example in cancer, population genetics, sampling theory and ecology. In this paper we explore properties of one model that is embedded into a continuous-time process and can describe the appearance of certain biological data such as covid DNA sequences in a database. More specifically, we consider an evolving model of counts-of-counts data that arises as the family size counts of samples taken sequentially from a Birth process with Immigration (BI). Here, each family represents a type or species, and the family size counts represent the type or species frequency spectrum in the population. We study the correlation of $S(a,b)$ and $S(c,d)$, the number of families observed in two disjoint time intervals $(a,b)$ and $(c,d)$. We find the expected sample variance and its asymptotics for $p$ consecutive sequential samples $\mathbf{S}_p:=(S(t_0,t_1),\dots, S(t_{p-1},t_p))$, for any given $0=t_0<t_1<\dots<t_p$. By conditioning on the sizes of the samples, we provide a connection between $\mathbf{S}_p$ and $p$ sequential samples of sizes $n_1,n_2,\dots,n_p$, drawn from a single run of a Chinese Restaurant Process. The properties of the latter were studied in da Silva et al. (2022). We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of $t_1,t_2,\dots, t_p$, we revisit Fisher's 1943 multi-sampling problem and give another explanation of what Fisher's model could have meant in the world of sequential samples drawn from a BI process.