Another view of sequential sampling in the birth process with immigration

Abstract 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 $$S_p =(S(t_0,t_1),\dots , S(t_{p-1},t_p))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , for $$0=t_0&lt;t_1&lt;\dots &lt;t_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> . By conditioning on the sizes of the samples, we provide a connection between $$S_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> and p sequential samples of sizes $$n_1,n_2,\dots ,n_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> , 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 $$t_1,t_2,\dots , t_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> , 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.