Asymptotic properties of Bernstein estimators on the simplex

Abstract

In this paper, we study various asymptotic properties (bias, variance, mean squared error, mean integrated squared error, 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 (2012a) and Babu et al. (2002), which treated the case d = 1, and significantly extend those found in Tenbusch (1994) for the density estimators when d = 2. The density estimator (or smoothed histogram) is closely related to the Dirichlet kernel estimator from Ouimet (2020a), and can also be used to analyze compositional data.

Moments of the Negative Multinomial Distribution

“…Using the moment formulas in the present paper, study the asymptotic properties of the Bernstein estimator with a negative multinomial kernel, as was carried out for the Bernstein estimator with a multinomial kernel on the simplex in Ouimet [40].…”

A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

“…(In the setting of Bernstein estimators, c.d.f. estimation on compact sets was tackled, for example, by Babu et al [42], Leblanc [43], Leblanc [44], Leblanc [45], Dutta [46], Jmaei et al [47], Erdo gan et al [48] and Wang et al [49] in the univariate setting, and by Babu and Chaubey [50], Belalia [51], Dib et al [52] and Ouimet [53,54] in the multivariate setting. In [55], the authors introduced Bernstein estimators with Poisson weights (also called Szasz estimators) for the estimation of c.d.f.s that are supported on [0, ∞), see also Ouimet [56]).…”