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Bernstein polynomial distribution estimators and the Dvoretzky–Kiefer–Wolfowitz inequality

Volume 49 / 2022

Waldemar Popiński Applicationes Mathematicae 49 (2022), 175-184 MSC: Primary 62G30; Secondary 60E15. DOI: 10.4064/am2461-11-2022 Published online: 15 December 2022


The problem of nonparametric distribution function estimation using the Bernstein polynomials is investigated. It is proved that the Bernstein polynomial estimator that was most often examined so far does not satisfy the Dvoretzky–Kiefer–Wolfowitz inequality in the class of continuous distribution functions supported on $[0,1]^d$, $d\in \mathbb {N}$. However, for a family of equicontinuous distributions the Bernstein polynomial estimators do satisfy a DKW type inequality for sufficiently large polynomial degree.


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