On the boundary properties of Bernstein estimators on the simplex

Abstract

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. Our results generalize those found by Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (2021) in the interior of the simplex. Since the "edges" of the $d$-dimensional simplex have dimensions going from $0$ (vertices) up to $d-1$ (facets) and our kernel function is multinomial, the asymptotic expressions for the bias, variance and mean squared error are not straightforward extensions of one-dimensional asymptotics as they would be for product-type estimators studied by almost all past authors in the context of Bernstein estimators or asymmetric kernel estimators. This point makes the mathematical analysis much more interesting.

16 pages, 0 figures

References in corpus (4)

• Asymptotic properties of Dirichlet kernel density estimators
• Asymptotic properties of Bernstein estimators on the simplex
• A precise local limit theorem for the multinomial distribution and some applications
• General formulas for the central and non-central moments of the multinomial distribution

Cited by in corpus (5)

• Asymptotic properties of Dirichlet kernel density estimators
• Asymptotic properties of Bernstein estimators on the simplex
• A precise local limit theorem for the multinomial distribution and some applications
• Explicit formulas for the joint third and fourth central moments of the multinomial distribution
• Normal approximations for the multivariate inverse Gaussian distribution and asymmetric kernel smoothing on $d$-dimensional half-spaces