Asymptotic properties of Bernstein estimators on the simplex

Abstract

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

22 pages, 1 figure

References in corpus (7)

• Shape restricted regression with random Bernstein polynomials
• Asymptotic properties of Dirichlet kernel density estimators
• A precise local limit theorem for the multinomial distribution and some applications
• On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators
• Smooth Distribution Function Estimation for Lifetime Distributions using Szasz-Mirakyan Operators
• On the boundary properties of Bernstein estimators on the simplex
• Maximum Approximate Likelihood Estimation in Accelerated Failure Time Model for Interval-Censored Data

Cited by in corpus (10)

• Asymptotic properties of Dirichlet kernel density estimators
• 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
• On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators
• A study of seven asymmetric kernels for the estimation of cumulative distribution functions
• On the boundary properties of Bernstein estimators on the simplex
• Minimax properties of Dirichlet kernel density estimators
• Local linear smoothing for regression surfaces on the simplex using Dirichlet kernels
• Explicit formulas for the joint third and fourth central moments of the multinomial distribution
• Moments of the negative multinomial distribution