Maximum likelihood estimation for spinal-structured trees

TL;DR

This paper studies the challenges of estimating birth distributions in spinal-structured multi-type Galton-Watson trees, revealing a trade-off between population growth rate and distribution dissimilarity affecting estimation accuracy.

Contribution

It introduces a framework for maximum likelihood estimation in spinal-structured trees and characterizes the conditions under which estimation is feasible based on growth rate and distribution divergence.

Findings

Estimation is possible if growth rate is below a divergence threshold.

Large deviations hinder type distinction at high growth rates.

A divergence measure determines the feasibility of accurate estimation.

Abstract

We investigate some aspects of the problem of the estimation of birth distributions (BD) in multi-type Galton-Watson trees (MGW) with unobserved types. More precisely, we consider two-type MGW called spinal-structured trees. This kind of tree is characterized by a spine of special individuals whose BD $\nu$ is different from the other individuals in the tree (called normal whose BD is denoted $\mu$). In this work, we show that even in such a very structured two-type population, our ability to distinguish the two types and estimate $\mu$ and $\nu$ is constrained by a trade-off between the growth-rate of the population and the similarity of $\mu$ and $\nu$. Indeed, if the growth-rate is too large, large deviations events are likely to be observed in the sampling of the normal individuals preventing us to distinguish them from special ones. Roughly speaking, our approach succeeds if $r<\mathfrak{D}(\mu,\nu)$ where $r$ is the exponential growth-rate of the population and $\mathfrak{D}$ is a divergence measuring the dissimilarity between $\mu$ and $\nu$.