  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 phasty, and R code for the
reproduction of our results is available as an accompanying vignette.
