TMRCA is observable from the branch lengths of a coalescent
The height of ancestral trees, in other words the time to the most recent common ancestor in a genealogical tree, is an informative statistic in population genomics and used in contexts of genealogical dating, demographic history inference, and assessing signals of selection. Exploiting a general pathwise identity between the height of a tree and its branch lengths, we revisit Zeng et al.s mutation rate estimator to develop a model-free estimator for the product of the expected height of genealogical trees with the scaled mutation rate, both for single loci and genomic regions. This estimator is a linear function of the site frequency spectrum of a sample, i.e., of the distribution of allele counts at all segregating sites across the locus or genomic region and thus both observable and computationally cheap. We show that, under the infinite sites model of mutation, our estimator is unbiased and, for two genome-wide models of ancestries (sequential Markovian coalescents and common-pedigree ancestries), consistent. Furthermore, we show via simulation that our estimator performs smaller errors than averaging over reconstructed ancestral tree heights (extracted from reconstructed ancestral recombination graphs) when considering genome properties similar to human genomes across large genomic regions. We then revisit a publicly available genomic dataset of dogs and wolves and compare our method with extracting TMRCAs from ancestral recombination graph reconstructions.