On the boundary properties of Bernstein estimators on the simplex

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature. Our results generalize those found in Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (2020) in the interior of the simplex. Different parts of the boundary having different dimensions makes the analysis more difficult.