  Bernstein estimators are well-known to avoid the boundary bias problem of
traditional kernel estimators. The theoretical properties of these estimators
have been studied extensively on compact intervals and hypercubes, but never on
the simplex, except for the mean squared error of the density estimator in
Tenbusch (1994) when $d = 2$. The simplex is an important case as it is the
natural domain of compositional data. In this paper, we make an effort to prove
several asymptotic results (bias, variance, mean squared error (MSE), mean
integrated squared error (MISE), asymptotic normality, uniform strong
consistency) for Bernstein estimators of cumulative distribution functions and
density functions on the $d$-dimensional simplex. Our results generalize the
ones in Leblanc (2012) and Babu et al. (2002), who treated the case $d = 1$,
and significantly extend those found in Tenbusch (1994). In particular, our
rates of convergence for the MSE and MISE are optimal.
