On the boundary properties of Bernstein estimators on the simplex

TL;DR

This paper investigates the asymptotic bias, variance, and mean squared error of Bernstein estimators for distribution and density functions on the boundary of a d-dimensional simplex, extending previous work to higher dimensions.

Contribution

It generalizes boundary asymptotic properties of Bernstein estimators from one dimension to higher dimensions, accounting for complex boundary geometries.

Findings

Derived asymptotic expressions for bias, variance, and MSE near boundaries.

Extended previous one-dimensional results to multidimensional simplexes.

Provided mathematical analysis for multinomial kernel-based estimators.

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.