Application of Bernstein Polynomials on Estimating a Distribution and Density Function in a Triangular Array

In this paper, we study some asymptotic properties for the Bernstein estimators of the limit distribution function and the limit density function under a triangular sample. Specifically, we obtain the uniform strong consistency, mean squared error (MSE) and mean integrated squared error (MISE) for the resulting estimators. In addition, we give the optimal choice of the bandwidth parameter m in terms of the sample size n, for both the MSE and MISE. Numerical simulations are presented to show that the Bernstein estimators outperform Gaussian kernel estimators in terms of MISE under a triangular sample.