A non-zero variance of Tajima’s estimator for two sequences even for infinitely many unlinked loci

Abstract

The population-scaled mutation rate,{theta} , is informative on the effective population size and is thus widely used in population genetics. We show that for two sequences and n unlinked loci, Tajimas estimator ([Formula]), which is the average number of pairwise differences, is not consistent and therefore its variance does not vanish even as n [->] {infty}. The non-zero variance of [Formula] results from a (weak) correlation between coalescence times even at unlinked loci, which, in turn, is due to the underlying fixed pedigree shared by all genealogies. We derive the correlation coefficient under a diploid, discrete-time, Wright-Fisher model, and we also derive a simple, closed-form lower bound. We also obtain empirical estimates of the correlation of coalescence times under demographic models inspired by large-scale human genealogies. While the effect we de scribe is small [Formula], it is important to recognize this feature of statistical population genetics, which runs counter to commonly held notions about unlinked loci.