Ancestral Inference and Learning for Branching Processes in Random Environments

TL;DR

This paper develops a new generalized method of moments approach for ancestral inference in branching processes within random environments, revealing how environmental variability affects estimator behavior and demonstrating asymptotic independence of estimators.

Contribution

It introduces a novel inference methodology for branching processes in random environments and analyzes the asymptotic properties of estimators under environmental variability.

Findings

Estimator behavior depends on environment coefficient of variation

Joint estimators of ancestor and offspring means become asymptotically independent Gaussian variables

Provides estimators for the limiting variance and validates with numerical and real data

Abstract

Ancestral inference for branching processes in random environments involves determining the ancestor distribution parameters using the population sizes of descendant generations. In this paper, we introduce a new methodology for ancestral inference utilizing the generalized method of moments. We demonstrate that the estimator's behavior is critically influenced by the coefficient of variation of the environment sequence. Furthermore, despite the process's evolution being heavily dependent on the offspring means of various generations, we show that the joint limiting distribution of the ancestor and offspring estimators of the mean, under appropriate centering and scaling, decouple and converge to independent Gaussian random variables when the ratio of the number of generations to the logarithm of the number of replicates converges to zero. Additionally, we provide estimators for the limiting variance and illustrate our findings through numerical experiments and data from Polymerase Chain Reaction experiments and COVID-19 data.