A note on the performance of the gamma kernel estimators at the boundary

Abstract

The gamma kernel estimator is proposed in Chen [Chen, S.X., 2000. Probability density function estimation using gamma kernels. Annals of the Institute of Statistical Mathematics 52, 471–480] to estimate densities with support [ 0 , ∞ ) . It is shown in his paper that the gamma kernel estimator is non-negative, free of boundary bias, and achieves the optimal rate of convergence for the mean integrated squared error. Numerical results reported in Chen’s paper show that, in the boundary region, the gamma kernel estimator even outperforms some widely used boundary corrected density estimators such as the boundary kernel estimator. However, our study finds that the gamma kernel estimator at x = 0 is actually the reflection estimator when the double exponential kernel is used and is only boundary problem free when the estimated density has a shoulder at x = 0 (i.e., the first derivative of the density at x = 0 is zero). For densities not satisfying the shoulder condition, we show that the gamma kernel estimator has a severe boundary problem and its performance is inferior to that of the boundary kernel estimator.