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

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.