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 Sp=(S(t0,t1),⋯,S(tp-1,tp))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_p =(S(t_0,t_1),\\dots , S(t_{p-1},t_p))$$\\end{document}, for 0=t0<t1<⋯<tp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0=t_0<t_1<\\dots <t_p$$\\end{document}. By conditioning on the sizes of the samples, we provide a connection between Sp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_p$$\\end{document} and p sequential samples of sizes n1,n2,⋯,np\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1,n_2,\\dots ,n_p$$\\end{document}, 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_1,t_2,\\dots , t_p$$\\end{document}, 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.