On the boundary properties of Bernstein polynomial estimators of density and distribution functions

Abstract

For density and distribution functions supported on [0,1], Bernstein polynomial estimators are known to have optimal mean integrated squared error (MISE) properties under the usual smoothness conditions on the function to be estimated. These estimators are also known to be well-behaved in terms of bias: they have uniform bias over the entire unit interval. What is less known, however, is that some of these estimators do experience a boundary effect, but of a different nature than what is seen with the usual kernel estimators.In this note, we examine the boundary properties of Bernstein estimators of density and distribution functions. Specifically, we show that Bernstein density estimators have decreased bias, but increased variance in the boundary region. In the case of distribution function estimation, we show that Bernstein estimators experience an advantageous boundary effect. Indeed, we prove a particularly impressive property of Bernstein distribution function estimators: they have decreased bias and variance in the boundary region. Finally, we also pay attention to the impact of the so-called shoulder condition on the boundary behaviour of these estimators.