Asymptotic properties of Bernstein estimators on the simplex

Frédéric Ouimet

Journal of Multivariate Analysis, 2021, vol. 185, issue C

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 (2012a) 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.

Keywords: Asymptotic normality; Bernstein estimators; Compositional data; Cumulative distribution function estimation; Density estimation; Mean squared error; Simplex; Uniform strong consistency (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.jmva.2021.104784

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