  The local polynomial density (LPD) estimator has been a useful tool for
inference concerning boundary points of density functions. While it is commonly
believed that kernel selection is not crucial for the performance of
kernel-based estimators, this paper argues that this does not hold true for LPD
estimators at boundary points. We find that the commonly used kernels with
compact support lead to larger asymptotic and finite-sample variances.
Furthermore, we present theoretical and numerical evidence showing that such
unfavorable variance properties negatively affect the performance of
manipulation testing in regression discontinuity designs, which typically
suffer from low power. Notably, we demonstrate that these issues of increased
variance and reduced power can be significantly improved just by using a kernel
function with unbounded support. We recommend the use of the spline-type kernel
(the Laplace density) and illustrate its superior performance.
