Asymptotic properties of Bernstein estimators on the simplex

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