Asymptotic properties of Bernstein estimators on the simplex

02/18/2020

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by   Frédéric Ouimet, et al.

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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 (2012) 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 (2020), and can also be used to analyze compositional data.