Beta kernel estimators for density functions

Abstract

Kernel estimators using non-negative kernels are considered to estimate probability density functions with compact supports. The kernels are chosen from a family of beta densities. The beta kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The proposed beta kernel estimators have two features. One is that the different amount of smoothing is allocated by naturally varying kernel shape without explicitly changing the value of the smoothing bandwidth. Another feature is that the support of the beta kernels can match the support of the density function; this leads to larger effective sample sizes used in the density estimation and can produce density estimates that have smaller finite-sample variance than some other estimators.