Consistency and Central Limit Results for the Maximum Likelihood Estimator in the Admixture Model

TL;DR

This paper establishes the consistency and asymptotic normality of maximum likelihood estimators in the Admixture Model, providing theoretical foundations and quantifying uncertainty in genetic ancestry and allele frequency estimates.

Contribution

It proves the consistency and central limit theorems for MLEs in the Admixture Model, including boundary cases, advancing statistical understanding of ancestry estimation.

Findings

Proved consistency of MLEs for allele frequencies and ancestries.

Established central limit theorems for finite samples and markers.

Quantified uncertainty of MLEs using real data.

Abstract

In the Admixture Model, the probability of an individual having a certain number of alleles at a specific marker depends on the allele frequencies in $K$ ancestral populations and the fraction of the individual's genome originating from these ancestral populations. This study investigates consistency and central limit results of maximum likelihood estimators (MLEs) for the ancestry and the allele frequencies in the Admixture Model, complimenting previous work by \cite{pfaff2004information, pfaffelhuber2022central}. Specifically, we prove consistency of the MLE, if we estimate the allele frequencies and the ancestries. Furthermore, we prove central limit theorems, if we estimate the ancestry of a finite number of individuals and the allele frequencies of finitely many markers, also addressing the case where the true ancestry lies on the boundary of the parameter space. Finally, we use the new theory to quantify the uncertainty of the MLEs for the data of \citet{10002015global}.