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：考虑一个在参数族后代分布下的多维超临界分支过程。在此过程中，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：即抽取随机样本，每个个体报告其子代规模向量。本文证明，在满足特定抽样规模条件时，样本中无同胞被同时抽样（从而使样本可视为独立）的概率趋于1。进一步，我们证明观测规模的抽样分布收敛于同分布乘积形式，由此建立框架使该过程可视为独立同分布，从而适用常规参数估计方法。我们给出了所得估计量的渐近分布。