A non-zero variance of Tajima's estimator for two sequences even for infinitely many unlinked loci
The population-scaled mutation rate, Θ, 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, Tajima's estimator (Θ^), which is the average number of pairwise differences, is not consistent and therefore its variance does not vanish even as n → ∞. The non-zero variance of Θ^ 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 describe is small (Var[Θ^]/Θ2 ≈ O(N-1)), it is important to recognize this feature of statistical population genetics, which runs counter to commonly held notions about unlinked loci.
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