Sampling distributions and estimation for multi-type branching processes

This study focuses on a multi-dimensional supercritical branching process with offspring distribution in a parametric family. Each vector coordinate represents the number of offspring of a given type, and the process is observed under family-size sampling: a random sample is drawn from the population, and each individual reports its vector of brood sizes. We show that the probability of no siblings being sampled (so that the sample can be considered independent) converges to one under specific conditions on the sample size. Moreover, the sampling distribution of the observations converges to a weighted mixture of offspring distributions, enabling observation to be considered an i.i.d. sample of a mixture law for which standard inference methodology applies. We provide asymptotic distributions for the resulting estimators and conduct a simulation study using respondent-driven sampling to assess their performance.