Another view of sequential sampling in the birth process with immigration

We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. Letting S ( a , b ) be the number of families observed in ( a , b ), we study the expected sample variance and its asymptotics for p consecutive sequential samples Sₚ =(S(t₀,t₁), , S(tₚ₋₁,tₚ)) S p = ( S ( t 0 , t 1 ) , ⋯ , S ( t p - 1 , t p ) ) , for 0=t₀<t₁< <tₚ 0 = t 0 < t 1 < ⋯ < t p . By conditioning on the sizes of the samples, we provide a connection between Sₚ S p and p sequential samples of sizes n₁,n₂, ,nₚ n 1 , n 2 , ⋯ , n p , drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166–1194, 2023. https://doi.org/10.3150/22-BEJ1494 ). 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 t₁,t₂, , tₚ t 1 , t 2 , ⋯ , t p , where the lengths of intervals are logarithmically equal, 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.