  Compositional data, which is data consisting of fractions or probabilities,
is common in many fields including ecology, economics, physical science and
political science. If these data would otherwise be normally distributed, their
spread can be conveniently represented by a multivariate normal distribution
truncated to the non-negative space under a unit simplex. Here this
distribution is called the simplex-truncated multivariate normal distribution.
For calculations on truncated distributions, it is often useful to obtain rapid
estimates of their integral, mean and covariance; these quantities
characterising the truncated distribution will generally possess different
values to the corresponding non-truncated distribution. In this paper, three
different approaches that can estimate the integral, mean and covariance of any
simplex-truncated multivariate normal distribution are described and compared.
These three approaches are (1) naive rejection sampling, (2) a method described
by Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Ross
algorithm with an analytical version of elliptical slice sampling, and (3) a
semi-analytical method that expresses the integral, mean and covariance in
terms of integrals of hyperrectangularly-truncated multivariate normal
distributions, the latter of which are readily computed in modern mathematical
and statistical packages. Strong agreement is demonstrated between all three
approaches, but the most computationally efficient approach depends strongly
both on implementation details and the dimension of the simplex-truncated
multivariate normal distribution. For computations in low-dimensional
distributions, the semi-analytical method is fast and thus should be
considered. As the dimension increases, the Gessner et al. method becomes the
only practically efficient approach of the methods tested here.
