关于有移民的纯生过程中序贯抽样的另一种看法
Another view of sequential sampling in the birth process with immigration
摘要：
计数的计数数据的模型已被广泛用于生物科学，例如癌症，群体遗传学，抽样理论和生态学。在本文中，我们探索了一个模型的属性，该模型嵌入到连续时间过程中，可以描述数据库中某些生物数据（如COVID DNA序列）的外观。更具体地说，我们认为一个不断发展的模型的计数的计数数据所产生的家庭规模计数的样本顺序从移民（BI）的纯生过程。在这里，每个家庭代表一个类型或物种，家庭大小计数代表种群中的类型或物种频谱。我们研究了$S(a,b)$和$S(c,d)$的相关性，以及在两个不相交的时间间隔$(a,b)$和$(c,d)$中观察到的家族数量。对于任何给定的$0=t_0<t_1<\dots<t_p$，我们发现$p$连续序列样本$\mathbf{S}_p:=(S(t_0,t_1),\dots, S(t_{p-1},t_p))$的预期样本方差及其渐近性。通过调节样本的大小，我们提供了大小为$n_1,n_2,\dots,n_p$的$\mathbf{S}_p$和$p$顺序样本之间的连接，这些样本来自中国餐馆过程的单次运行。da Silva等人（2022）研究了后者的性质。我们展示了连续时间框架如何有助于使渐近计算比离散时间更容易。作为一个应用程序，对于一个特定的选择$t_1,t_2,\dots, t_p$，我们重温费舍尔1943年的多抽样问题，并给出另一种解释费舍尔的模型可能意味着在世界上的顺序样本从BI过程。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
