Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval

Abstract

This paper demonstrates that two classes of multiplicative bias correction (MBC) techniques, originally proposed for density estimation using symmetric second-order kernels by Terrell and Scott (1980) and Jones et al. (1995), can be applied to density estimation using the beta and modified beta kernels. It is shown that, under sufficient smoothness of the true density, both MBC techniques reduce the order of magnitude in bias, whereas the order of magnitude in variance remains unchanged. Accordingly, mean squared errors of these MBC estimators achieve a faster convergence rate of O ( n − 8 / 9 ) for the interior part, when best implemented. Furthermore, the estimators always generate nonnegative density estimates by construction. To implement the MBC estimators, a plug-in smoothing parameter choice method is proposed. Monte Carlo simulations indicate good finite sample performance of the estimators.