Multivariate phase-type theory for the site frequency spectrum

Linear functions of the site frequency spectrum (SFS) play a major role for understanding and investigating genetic diversity. Estimators of the mutation rate (e.g. based on the total number of segregating sites or average of the pairwise differences) and tests for neutrality (e.g. Tajima's D) are perhaps the most well-known examples. The distribution of linear functions of the SFS is important for constructing confidence intervals for the estimators, and to determine significance thresholds for neutrality tests. These distributions are often approximated using simulation procedures. In this paper we use multivariate phase-type theory to specify, characterize and calculate the distribution of linear functions of the site frequency spectrum. In particular, we show that many of the classical estimators of the mutation rate are distributed according to a discrete phase-type distribution. Neutrality tests, however, are generally not discrete phase-type distributed. For neutrality tests we derive the probability generating function using continuous multivariate phase-type theory, and numerically invert the function to obtain the distribution. A main result is an analytically tractable formula for the probability generating function of the SFS. Software implementation of the phase-type methodology is available in the R package PhaseTypeR, and R code for the reproduction of our results is available as an accompanying vignette.