On improving convergence rate of Bernstein polynomial density estimator

Abstract

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, in Statistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’, The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’, Journal of Nonparametric Statistics, 22, 459–475] are shown to be O(n−8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O(n−8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.