In the Admixture Model, the probability that an individual carries a certain allele at a specific marker depends on the allele frequencies in $K$ ancestral populations and the proportion of the individual's genome originating from these populations. The markers are assumed to be independent. The Linkage Model is a Hidden Markov Model (HMM) that extends the Admixture Model by incorporating linkage between neighboring loci. We prove consistency and asymptotic normality of maximum likelihood estimators (MLEs) for the ancestry of individuals in the Linkage Model, complementing earlier results by \citep{pfaff2004information, pfaffelhuber2022central, HEINZEL2025} for the Admixture Model. These results are used to prove that a statistical test that allows for model selection between the Admixture Model and the Linkage Model is an asymptotic level-$α$-test. Finally, we demonstrate the practical relevance of our results by applying the test to real-world data from the 1000 Genomes Project.