Consider a multi-dimensional supercritical branching process with offspring distribution in a parametric family. Here, each vector coordinate corresponds to the number of offspring of a given type. The process is observed under family-size sampling: a random sample is drawn, each individual reporting its vector of brood sizes. In this work, we show that the set in which no siblings are sampled (so that the sample can be considered independent) has probability converging to one under certain conditions on the sampling size. Furthermore, we show that the sampling distribution of the observed sizes converges to the product of identical distributions, hence developing a framework for which the process can be considered iid, and the usual methods for parameter estimation apply. We provide asymptotic distributions for the resulting estimators.

翻译：考虑一个具有参数化族中后代分布的多维超临界分支过程。其中，每个向量坐标对应给定类型的后代数量。该过程在家庭规模抽样下被观测：随机抽取一个样本，每个个体报告其育雏规模的向量。本研究表明，在抽样规模满足特定条件时，未抽样到任何同胞个体（因而样本可被视为独立）的事件集合概率收敛于一。进一步，我们证明观测规模的抽样分布收敛于相同分布的乘积，从而建立了一个可将该过程视为独立同分布的理论框架，使得常规参数估计方法得以适用。我们给出了所得估计量的渐近分布。